Lenses, devices and methods for ocular refractive error

ABSTRACT

A lens for an eye having an optical axis and an aberration profile along its optical axis, the aberration profile having a focal distance and including higher order aberrations having at least one of a primary spherical aberration component and a secondary spherical aberration component. The aberration profile may provide, for a model eye with no aberrations and an on-axis length equal to the focal distance: a peak, first retinal image quality (RIQ) within a through focus range that remains at or above a second RIQ over the through focus range that includes said focal distance, where the first RIQ is at least 0.35, the second RIQ is at least 0.1 and the through focus range is at least 1.8 Diopters.

CROSS REFERENCE TO RELATED MATERIALS

This application is a continuation of U.S. patent application Ser. No.16/226,263, filed Dec. 19, 2018, which is a continuation of U.S. patentapplication Ser. No. 15/399,445, filed Jan. 5, 2017, now U.S. Pat. No.10,209,535 issued Feb. 19, 2019, which is a continuation of U.S.application Ser. No. 14/565,062, filed Dec. 9, 2014, now U.S. Pat. No.9,575,334 issued Feb. 21, 2017, which is a continuation of U.S.application Ser. No. 13/857,613, filed Apr. 5, 2013, now U.S. Pat. No.9,195,074, issued Nov. 24, 2015, which claims priority to AustralianProvisional Application No. 2012/901382, entitled, “Devices and Methodsfor Refractive Error Control” filed on Apr. 5, 2012, and AustralianProvisional Application No. 2012/904541 entitled “Lenses, Devices andMethods for Ocular Refractive Error”, filed on Oct. 17, 2012. Each ofthese applications are incorporated herein by reference in theirentirety.

FIELD

The disclosed embodiments include lenses, devices and methods forchanging or controlling the wavefront of light entering an eye, inparticular a human eye.

Particular disclosed embodiments include lenses, devices and methods formitigating/addressing ocular refractive error, in particular in humaneyes. The ocular refractive error may for example arise from myopia orhyperopia, with or without astigmatism. The ocular refractive error mayarise from presbyopia, either alone or in combination with myopia orhyperopia and with or without astigmatism.

The disclosed embodiments of lenses, devices and methods includeembodiments that address foveal vision and embodiments that address bothfoveal and peripheral vision.

Examples of lenses in the fields of the disclosed embodiments includecontact lenses, corneal onlays, corneal inlays, and lenses forintraocular devices (both anterior and posterior chamber).

Examples of devices in the fields of the disclosed embodiments includeaccommodating intraocular lenses and electro-active spectacle lenses.

Examples of methods in the fields of the disclosed embodiments includemethods of changing the refractive state/wavefront of light entering aneye and received by a retina of the eye (e.g. refractive surgery,corneal ablation), methods of design and/or manufacture of lenses andoptical devices, methods of surgery to alter the refractive state of aneye and methods of controlling stimulus for progression of eye growth.

BACKGROUND

For an image to be perceived clearly, the optics of the eye shouldresult in an image that is focused on the retina. Myopia, commonly knownas short-sightedness, is an optical disorder of the eye wherein on-axisimages are focused in front of the fovea of the retina. Hyperopia,commonly known as long-sightedness, is an optical disorder of the eyewherein on-axis images are focused behind the fovea of the retina. Thefocusing of images in front of or behind the fovea of the retina createsa lower order aberration of defocus. Another lower order aberration isastigmatism. An eye may also have higher order optical aberrations,including for example spherical aberration, coma and/or trefoil. Manypeople experiencing natural refractive error are progressing (therefractive error is increasing over time). Progression is particularlywidespread in people with myopia. Schematic representations of eyesexhibiting myopia or hyperopia and astigmatism are shown in FIGS. 1A-Crespectively. In a myopic eye 100, the parallel incoming beam of light102 passes the refractive elements of the eye, namely, the cornea 104and crystalline lens 106, to a focal point 108 short of the retina 110.The image on the retina 110 is therefore blurred. In a hyperopic eye120, the parallel incoming beam of light 122 passes the refractiveelements of the eye, namely, the cornea 124 and crystalline lens 126, toa focal point 128 beyond the retina 130, again rendering the image onthe retina 130 blurred. In an astigmatic eye 140, the parallel incomingbeam of light 142 passes the refractive elements of the eye, namely,cornea 144 and crystalline lens 146, and results in two foci, namelytangential 148 and sagital 158 foci. In the example of astigmatism shownin FIG. 1C, the tangential focus 148 is in front the retina 160 whilethe sagital focus 158 is behind the retina 160. The image on the retinain the astigmatic case is referred to as circle of least confusion 160.

At birth human eyes are hyperopic, i.e. the axial length of the eyeballis too short for its optical power. With age, from infancy to adulthood,the eyeball continues to grow until its refractive state stabilizes.Elongation of the eye in a growing human may be controlled by a feedbackmechanism, known as the emmetropisation process, so that the position offocus relative to the retina plays a role in controlling the extent ofeye growth. Deviation from this process would potentially result inrefractive disorders like myopia, hyperopia and/or astigmatism. Whilethere is ongoing research into the cause of deviation of emmetropisationfrom stabilising at emmetropia, one theory is that optical feedback canprovide a part in controlling eye growth. For example, FIG. 2 showscases that would, under a feedback mechanism theory of theemmetropisation process, alter the emmetropisation process. In FIG. 2A,the parallel incoming beam of light 202 passes through a negativerefractive element 203 and the refractive elements of the eye (thecornea 204 and crystalline lens 206), to form an image at focus point208, overshooting the retina 210. The resulting image blur on theretina, called hyperopic defocus, is an example of defocus that mayencourage eye growth under this feedback mechanism. In contrast, as seenin FIG. 2B, the parallel incoming beam of light 252 passes through apositive refractive element 253, the refractive elements of the eye(cornea 254 and crystalline lens 256) to form an image at focus point258 in front of the retina 260. The resulting image blur, called myopicdefocus, on this retina is considered to be an example of defocusinduced at the retina that would not encourage eye growth. Therefore, ithas been proposed that progression of myopic refractive error can becontrolled by positioning of the focus in front of the retina. For anastigmatic system, the spherical equivalent, i.e. the mid-point betweenthe tangential and sagital foci, may be positioned in front of theretina. These proposals have not however provided a full explanation orsolution, particularly in the case of progressing myopia.

A number of optical device designs and refractive surgery methods havebeen proposed to control the growth of the eye during emmetropisation.Many are generally based on refinements to the idea summarised abovethat foveal imagery provides a stimulus that controls the growth of theeye. In humans, the eye grows longer during emmetropisation and can notgrow shorter. Accordingly, during emmetropisation an eye may grow longerto correct for hyperopia, but it can not grow shorter to correct formyopia. Many proposals have been made for addressing myopia progression,some of which are summarised below.

U.S. Pat. No. 6,752,499 (Aller) proposes the use of bifocal contactlenses for myopic participants who exhibit near-point esophoria, forproviding a stimulus for reducing/controlling myopia progression. U.S.Pat. No. 7,025,460 (Smith et al) proposes the use of corrective eyelenses that shift the focal plane in front of the peripheral retina.U.S. Pat. No. 7,506,983 (To et al) proposes a method of treating myopiaprogression in human eyes by producing a secondary myopic image by useof Fresnel optics, while correcting the myopia of the candidate via therefractive portion of the lens. U.S. Pat. No. 7,997,725 (Phillips)proposes a method of simultaneous vision, wherein one part of thecorrecting lens corrects for pre-existing myopia while another part hasless negative power than the focal power of the lens to be able toproduce simultaneous myopic defocus and thereby aid in the retardationof myopia progression. U.S. Pat. No. 6,045,578 (Collins and Wildsoet)proposes the addition of positive spherical aberration at the fovea toprovide a stimulus that will reduce and/or control myopia progression.U.S. Pat. No. 7,401,922 (Legerton et al) proposes a method and system oftreating myopia progression in myopic patients by inducing certainaberration profiles that have positive spherical aberration to produce awavefront disposed in front of the retina. U.S. Pat. No. 7,803,153, B2(Thorn et al) proposes a method of preventing myopia progression throughidentification and correction of all optical aberrations, includinghigher order aberrations.

In addition to proposed optical strategies to counter the development ofrefractive error and its progression, in particular myopia, there hasalso been interest in strategies that involve non-optical interventionlike pharmacological substances, such as atropine or pirenzipine.

Another condition of the eye is presbyopia, in which the eye's abilityto accommodate is reduced or the eye has lost its ability toaccommodate. Presbyopia may be experienced in combination with myopia,hyperopia, astigmatism and higher order aberrations. Many differentmethods, devices and lenses to address presbyopia have been proposed,including in the form of bifocal, multifocal or progressive additionlenses/devices, which simultaneously provide two or more foci to theeye. Three common types of lenses used for presbyopia are centre-near,centre-distance aspheric multifocals and concentric (ring-type) bifocalsalternating between distance and near powers.

In addition, on occasion it is necessary to remove the crystalline lensof an eye, for example if the person is suffering from cataracts. Theremoved natural crystalline lens may be replaced by an intraocular lens.Accommodating intraocular lenses allow the eye to control the refractivepower of the lens, for example through haptics extending from the lensto the ciliary body.

SUMMARY

Disclosed herein are various lenses, devices and methods for providingan aberration profile for an eye. Characteristics of aberration profilesand methodologies for identifying aberration profiles are described formyopic eyes, hyperopic eyes and presbyopic eyes. In addition lenses,devices and methods for an eye with astigmatism are disclosed.

In one embodiment, a lens for an eye has an optical axis and anaberration profile about its optical axis, the aberration profile havinga focal distance and including at least one of a primary sphericalaberration component (C(4,0)) and a secondary spherical aberrationcomponent (C(6,0)). The aberration profile provides a retinal imagequality (RIQ) with a through focus slope that degrades in a direction ofeye growth; and a RIQ of at least 0.30. The RIQ is Visual Strehl Ratiomeasured along the optical axis for at least one pupil diameter in therange 3 mm to 6 mm, over a spatial frequency range of 0 to 30cycles/degree inclusive and at a wavelength selected from within therange 540 nm to 590 nm inclusive. In other embodiments the RIQ measuremay be different.

In another embodiment, a lens includes an optical axis and an aberrationprofile about the optical axis that provides a focal distance for aC(2,0) Zernike coefficient term; a peak Visual Strehl Ratio (‘firstVisual Strehl Ratio’) within a through focus range, and a Visual StrehlRatio that remains at or above a second Visual Strehl Ratio over thethrough focus range that includes said focal distance, wherein theVisual Strehl Ratio is measured for at least one pupil diameter in therange 3 mm to 5 mm, over a spatial frequency range of 0 to 30cycles/degree inclusive, at a wavelength selected from within the range540 nm to 590 nm inclusive, and wherein the first Visual Strehl Ratio isat least 0.35, the second Visual Strehl Ratio is at least 0.10 and thethrough focus range is at least 1.8 Diopters.

In one embodiment, a method for a presbyopic eye includes identifying awavefront aberration profile for the eye, the wavefront aberrationprofile including at least one spherical aberration term. Theprescription focal distance of the aberration profile is determinedtaking into account said spherical aberration and wherein theprescription focal distance is at least +0.25 D relative to a focaldistance for a C(2,0) Zernike coefficient term of the wavefrontaberration profile. The method includes producing a device, lens orcorneal profile for the eye to affect said wavefront aberration profile.

In one embodiment, a method for a myopic eye includes identifying awavefront aberration profile for the eye and applying or prescribing theaberration profile. The wavefront aberration profile includes at leastone spherical aberration term, wherein the prescription focal distanceof the aberration profile is determined taking into account saidspherical aberration and wherein the prescription focal distance is atleast +0.10 D relative to a focal distance for a C(2,0) Zernikecoefficient term of the wavefront aberration profile. The wavefrontaberration profile also provides a degrading retinal image quality inthe direction posterior to the retina.

A method for a hyperopic eye, the method comprising identifying awavefront aberration profile for the eye and applying or prescribing theaberration profile. The wavefront aberration profile includes at leastone spherical aberration term, wherein the prescription focal distanceof the wavefront aberration profile is determined taking into accountsaid spherical aberration. At the prescription focal distance thewavefront aberration profile provides an improving retinal image qualityin the direction posterior to the retina.

In some embodiments a computational device includes an input to receivefirst combination of aberrations, one or more processors to compute asecond combination of aberrations for one or more optical surfaces, andan output to output the second combination of aberrations, wherein thecomputed second combination of aberrations provides in combination withthe first combination of aberrations a total combination of HOA asdescribed above.

Further embodiments will become apparent from the following description,given by way of example and with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1C are schematic representations of eyes exhibiting myopia,hyperopia and astigmatism respectively.

FIGS. 2A and 2B are schematic representations respectively of hyperopicdefocus and myopic defocus induced at the retina.

FIG. 3 shows a two-dimensional through-focus point spread functioncomputed at the retinal plane without higher order aberrations (HOA) andin the presence of HOA of spherical aberration, vertical coma andhorizontal trefoil.

FIGS. 4 to 7 show graphs of the interaction of primary sphericalaberration with horizontal coma, vertical coma, horizontal trefoil andvertical trefoil respectively.

FIG. 8 shows a graph indicating the magnitude of myopia progressionunder an optical feedback mechanism for eye growth, for primaryspherical aberration vs. primary vertical astigmatism vs. primaryhorizontal astigmatism.

FIG. 9 shows a graph indicating the magnitude of myopia progression forprimary spherical aberration vs. secondary vertical astigmatism vs.secondary horizontal astigmatism.

FIG. 10 shows a graph indicating the myopia progression on a binaryscale for primary spherical aberration vs. secondary sphericalaberration.

FIG. 11 shows a graph indicating the myopia progression on a binaryscale for primary spherical aberration vs. tertiary sphericalaberration.

FIG. 12 shows a graph indicating the myopia progression on a binaryscale for primary spherical aberration vs. quaternary sphericalaberration.

FIG. 13 shows a graph indicating the myopia progression on a binaryscale for primary spherical aberration vs. secondary sphericalaberration vs. tertiary spherical aberration.

FIG. 14 shows example designs of aberration profiles that providenegative and positive gradient RIQ in a direction of eye growth.

FIG. 15 shows a work flow chart for myopic eyes, progressing ornon-progressing.

FIG. 16 shows a work flow chart for hyperopic eyes, progressing ornon-progressing towards emmetropia.

FIGS. 17 to 25 show example designs of power profiles of correcting lensacross the optic zone diameter, for affecting optical feedbackmechanisms for myopia.

FIG. 26 shows an example design of a power profile of correcting lensacross the optic zone diameter, for affecting optical feedbackmechanisms for hyperopia.

FIG. 27 shows a global through-focus retinal image quality (Q) for anaberration profile corresponding to a single vision lens.

FIG. 28 shows a global through-focus retinal image quality for a firstaberration profile (Iteration A1), which may have application to aprogressing myopic eye, and FIG. 29 shows the power profile for a lensfor providing the first aberration profile.

FIG. 30 shows a global through-focus retinal image quality for a secondaberration profile (Iteration A2), which may also have application to aprogressing myopic eye.

FIGS. 31 and 32 show a global through-focus retinal image quality for athird and fourth aberration profile (Iteration C1 and Iteration C2),which may have application to a hyperopic eye.

FIG. 33 show a retinal image quality (Q) for seven aberration profilesover a through focus range of 2.5D. The seven aberration profilescorrespond to example centre-distance and centre-near asphericmultifocals and concentric ring/annulus type bifocals and threeexemplary aberration profiles (Iteration B1, Iteration B2, Iteration B3)for optimising through focus performance.

FIGS. 34 to 40 show the power profiles of contact lenses across theoptic zone diameter, for providing the aberration profiles of FIG. 33 .

FIGS. 41 to 43 show the on-axis through-focus image quality for thethree exemplary embodiments for presbyopia (Iteration B1, B2 and B3)across four pupil diameters (3 mm to 6 mm) and FIGS. 44 and 45 show theon-axis through-focus image quality for the centre-distance andcentre-near aspheric multifocal designs across four pupil diameters (3mm to 6 mm).

FIGS. 46 and 47 show a monocular correction approach for presbyopia,where different higher order aberration profiles provided for the rightand left eyes, by which the through-focus optical/visual performance isdifferent in each eye (desired vergences) to provide a combined addpower range of 1.50D and 2.50D, on the negative side of through-focuscurve, respectively.

FIGS. 48 and 49 show a monocular correction approach for presbyopia,where different higher order aberration profiles provided for the rightand left eyes, by which the through-focus optical/visual performance isdifferent in each eye (desired vergences) to provide a combined addpower range of 1.50D and 2.50D, on the positive side of through-focuscurve, respectively.

FIG. 50 shows a global through-focus retinal image quality (Q) for threefurther iterations of aberration profile (Iterations A3, A4 and A5), forproviding a relatively constant retinal image quality across ahorizontal visual field from 0 to 30 degrees.

FIGS. 51 and 52 show example designs of the power profile of correctingcontact lenses with opposite phase profiles (Iteration E1 and IterationE2) and FIGS. 53 to 55 show the on-axis through-focus retinal imagequality (Q) for Iterations E1 and E2 with three different levels ofinherent primary spherical aberration.

FIG. 56 shows the through-focus RIQ performance measures (depth offocus) of 78 exemplary aberration profiles (Appendix A) that involve acombination of spherical aberration terms. The Y-axis in the graphdenotes ‘Q’ performance metric and X-axis denotes the through-focusrange from −1.50 to +1.00D. All the calculations were performed at 4 mmpupil. The solid black line indicates the through-focus performance of acombination that does not have any mode of spherical aberration whileall the gray lines indicate the 78 combinations which include at leastone higher order spherical aberration term. The 78 combinations wereselected with regard to performance on the negative side of thethrough-focus curve.

FIG. 57 shows the through-focus RIQ performance of one exemplarycombination from FIG. 56 that involves only positive sphericalaberration in comparison with a combination that has no sphericalaberration.

FIG. 58 shows the through-focus RIQ performance measures (depth offocus) of 67 exemplary aberration profiles that involve a combination ofspherical aberration terms (Appendix C). The Y-axis in the graph denotes‘Q’ performance metric and X-axis denotes the through-focus range from−1.50 to +1.00D. All the calculations were performed at 4 mm pupil. Thesolid black line indicates the through-focus performance of acombination that does not have any mode of spherical aberration whileall the gray lines indicate the 67 combinations which include at leastone higher order spherical aberration term. These 67 combinationsimprove performance on the positive side of the through-focus curve.

FIG. 59 shows a work flow chart for presbyopic eyes.

FIG. 60 shows a power profile for a tonic prescription of a contact lensfor both astigmatism and presbyopia.

FIG. 61 shows an example lens power profile, which is a combination ofspherical aberration terms and FIG. 62 shows the lens power profileconverted to an axial thickness profile for a contact lens.

DESCRIPTION OF EMBODIMENTS

The optical performance of the human eye is limited by a number offactors. Major factors include monochromatic and polychromatic opticalwavefront aberrations, in addition to the retinal sampling which imposesa Nyquist limit on spatial vision. Minor factors include theStiles-Crawford Effect and scattering. A quantitative measure of retinalimage quality (RIQ) can be defined once all factors affecting imagequality are quantified. A measure of RIQ may use a combination offactors that is a subset of all the factors that influence imagequality.

1. Retinal Image Quality (RIQ)

With use of a wavefront aberrometer, such as a Hartmann-Shackinstrument, the optical characteristics of a candidate eye can bemeasured so as to identify a measure of retinal image quality (RIQ).Several measures of RIQ are described below.

(A) Strehl Ratio

Once the wavefront aberration of the candidate eye is availed, the imagequality at the retina of the eye can be determined by computing thesimple Strehl ratio, as described in the Equation 1. The Strehl ratiocan be computed in both spatial domain (i.e. using Point spreadfunction) and in Fourier domain (i.e. using Optical transfer function asshown below in equation 1). The Strehl ratio measure is always boundbetween 0 and 1, where 1 is associated with best image quality.

$\begin{matrix}{{{{Strehl}'}s{ratio}} = \frac{\int{\int_{- \infty}^{+ \infty}\left( {FT}\left( ❘{FT}\left\{ {{A\left( {\rho,\theta} \right)} \star {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {W\left( {\rho,\theta} \right)}} \right\rbrack}} \right\} ❘^{2} \right) \right)}}{\int{\int_{- \infty}^{+ \infty}\left( {FT}\left( ❘{FT}\left\{ {{A\left( {\rho,\theta} \right)} \star {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {{Wdiff}\left( {\rho,\theta} \right)}} \right\rbrack}} \right\} ❘^{2} \right) \right)}}} & {{Equation}1}\end{matrix}$

(B) Visual Strehl Ratio

U.S. Pat. No. 7,077,522 B2 (Williams), which is hereby incorporated byreference herein in its entirety, describes a vision metric called thesharpness metric, by taking into account both the wavefront aberrationand the retinal response to the image. This metric can be computed byconvolving a point spread function with a neural quality function.Further, U.S. Pat. No. 7,357,509 B2 (Williams et al 2008) describesseveral other metrics to gauge optical performance of the human eye.

One such RIQ measure is the Visual Strehl Ratio, which calculated in thefrequency domain. The Visual Strehl Ratio in the frequency domain isdescribed by Equation 2 and is always bound between 0 and 1, where 1 isassociated with best image quality at the retina.

This metric only considers monochromatic aberrations.

$\begin{matrix}{{{monochromatic}{RIQ}} = \frac{\begin{matrix}{\int{\int_{- \infty}^{+ \infty}{{{CSF}\left( {f_{x},f_{y}} \right)} \star {{real}\left( {{FT}\left( {❘{{FT}\left\{ {{A\left( {\rho,\theta} \right)} \star} \right.}} \right.} \right.}}}} \\{\left. {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {W\left( {\rho,\theta} \right)}} \right\rbrack} \right\} ❘^{2}\text{))}}\end{matrix}}{\begin{matrix}{\int{\int_{- \infty}^{+ \infty}{{{CSF}\left( {f_{x},f_{y}} \right)}*\left( {{FT}\left( {❘{{FT}\left\{ {{A\left( {\rho,\theta} \right)}*} \right.}} \right.} \right.}}} \\{\left. {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {{Wdiff}\left( {\rho,\theta} \right)}} \right\rbrack} \right\} ❘^{2}\text{))}}\end{matrix}}} & {{Equation}2}\end{matrix}$

The RIQ measure of monochromatic Visual Strehl Ratio shows highcorrelation with objective and subjective visual acuity (e.g. Thibos etal. Journal of Vision 2004). This measure has been predominantly beenused to describe RIQ in the remainder of this specification. However,other measures described herein and alternatives thereto may be used inthe design of optical devices, lenses and methods.

(C) Polychromatic RIQ

The Visual Strehl Ratio defined by Williams, discussed above, is limitedto monochromatic light. To accommodate for polychromatic light, a metriccalled the polychromatic retinal image quality (polychromatic RIQ) isdefined that includes chromatic aberrations weighed with spectralsensitivities for selected wavelengths. The polychromatic RIQ measure isdefined in Equation 3.

$\begin{matrix}{{{polychromatic}{RIQ}} = \frac{\begin{matrix}{\int{\int_{- \infty}^{+ \infty}{{{CSF}\left( {f_{x},f_{y}} \right)} \star {\sum_{\lambda\min}^{\lambda\max}\left( {{S(\lambda)} \star} \right.}}}} \\{{real}\left( {FT}\left( ❘{FT}\left\{ {{A\left( {\rho,\theta} \right)} \star {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {W\left( {\rho,\theta} \right)}} \right\rbrack}} \right\} ❘^{2}\text{))))} \right. \right.}\end{matrix}}{\begin{matrix}{\int{\int_{- \infty}^{+ \infty}{{{CSF}\left( {f_{x},f_{y}} \right)}*{\sum_{\lambda\min}^{\lambda\max}\left( {{S(\lambda)} \star} \right.}}}} \\{\text{((}{FT}\left( ❘{FT}\left\{ {{A\left( {\rho,\theta} \right)}*{\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {{Wdiff}\left( {\rho,\theta} \right)}} \right\rbrack}} \right\} ❘^{2}\text{))))} \right.}\end{matrix}}} & {{Equation}3}\end{matrix}$

(D) Monochromatic Global RIQ

The Visual Strehl Ratio or monochromatic RIQ discussed above insub-section B is limited to on-axis vision. As used herein, unless thecontext clearly requires otherwise, ‘on-axis’ is a reference to any oneof the optical, visual or papillary axis. To accommodate for wide angleview (i.e. peripheral visual field), a metric called the global retinalimage quality (GRIQ) is defined that includes range of visual fieldeccentricities. A monochromatic GRIQ measure is defined in Equation 4.

$\begin{matrix}{{{monochromatic}{Global}{RIQ}} = \frac{\begin{matrix}{\int_{\alpha\min}^{\alpha\max}{\int_{\varphi\min}^{\varphi\max}\left\{ {\int{\int_{- \infty}^{+ \infty}{{{CSF}\left( {f_{x},f_{y}} \right)} \star {{real}\left( {FT}\left( {❘{{FT}\left\{ {{A\left( {\rho,\theta} \right)} \star} \right.}} \right. \right.}}}} \right.}} \\{\left. {\left. {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {W\left( {\rho,\theta} \right)}} \right\rbrack} \right\} ❘^{2}\text{))}} \right\}{d\varphi}d\lambda}\end{matrix}}{\begin{matrix}{\int_{\alpha\min}^{\alpha\max}{\int_{\varphi\min}^{\varphi\max}\left\{ {\int{\int_{- \infty}^{+ \infty}{{{CSF}\left( {f_{x},f_{y}} \right)}*}}} \right.}} \\{\left. \left( {FT}\left( ❘{FT}\left\{ {{A\left( {\rho,\theta} \right)}*{\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {W\left( {\rho,\theta} \right)}} \right\rbrack}} \right\} ❘^{2}\text{))} \right. \right. \right\}{d\varphi}d\lambda}\end{matrix}}} & {{Equation}4}\end{matrix}$

(E) Polychromatic Global RIQ

One other form of RIQ metric that accommodates for polychromatic lightand wide angle view (i.e. peripheral visual field), a metric is calledthe polychromatic global retinal image quality (GRIQ) is defined thatincludes chromatic aberrations weighed with spectral sensitivities forselected wavelengths and range of visual field eccentricities. Apolychromatic GRIQ measure is defined in Equation 5.

$\begin{matrix}{{{polychromatic}{Global}{RIQ}} = \frac{\begin{matrix}{\int_{\alpha\min}^{\alpha\max}{\int_{\varphi\min}^{\varphi\max}\left\{ {\int{\int_{- \infty}^{+ \infty}{{{CSF}\left( {f_{x},f_{y}} \right)} \star {\sum_{\lambda\min}^{\lambda\max}\left( {{S(\lambda)} \star \left( {{real}\left( {FT}\left( {❘{{FT}\left\{ {{A\left( {\rho,\theta} \right)} \star} \right.}} \right. \right.} \right.} \right.}}}} \right.}} \\{\left. {\left. {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {W\left( {\rho,\theta} \right)}} \right\rbrack} \right\} ❘^{2}\text{))))}} \right\}{d\varphi}d\lambda}\end{matrix}}{\begin{matrix}{\int_{\alpha\min}^{\alpha\max}{\int_{\varphi\min}^{\varphi\max}\left\{ {\int{\int_{- \infty}^{+ \infty}{{{CSF}\left( {f_{x},f_{y}} \right)}*{\sum_{\lambda\min}^{\lambda\max}\left( {{S(\lambda)} \star} \right.}}}} \right.}} \\{\left. \left( {FT}\left( ❘{FT}\left\{ {{A\left( {\rho,\theta} \right)}*{\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {{Wdiff}\left( {\rho,\theta} \right)}} \right\rbrack}} \right\} ❘^{2}\text{))))} \right. \right. \right\}{d\varphi}d\lambda}\end{matrix}}} & {{Equation}5}\end{matrix}$

In Equations 1 to 5:

-   -   f specifies the tested spatial frequency, this can be in the        range of F_(min) to F_(max) (denoting the boundary limits on the        spatial frequency content), for example F_(min)=0 cycles/degree;        F_(max)=30 cycles/degree;    -   f_(x) and f_(y) specifies the tested spatial frequency in x and        y directions;    -   CSF (f_(x), f_(y)) denotes a contrast sensitivity function,        which in a symmetric form can be defined as CSF        (F)=2.6(0.0192+0.114*f)*exp^(−(0.114*f){circumflex over ( )}1.1);        FT denotes, in one form of the equation, a 2D fast Fourier        transform; A(ρ, θ) and W(ρ, θ) denotes pupil diameter &        wavefront phase of the test case, respectively;

Wdiff (ρ, θ) denotes wavefront phase of the diffraction limited case;

ρ and θ are normalised polar coordinates, where p represents the radialcoordinate and

θ represents the angular coordinate or the azimuth;

λ denotes wavelength;

α denotes field angle;

φ denotes the meridian angle;

S(λ) denotes spectral sensitivity.

The wavefront phase, for example, can be written as a function set ofstandard Zernike polynomials up to a desired order, as described below,

${W\left( {\rho,\theta} \right)} = {\underset{i = 1}{\sum\limits^{k}}{a_{i}{Z_{i}\left( {\rho,\theta} \right)}}}$

Where, α_(i) denotes the i^(th) coefficient of Zernike polynomial

-   -   Z_(i)(ρ, θ), denotes the i^(th) Zernike polynomial term    -   ‘k’, represents the highest term of the expansion

These polynomials can be represented in the Optical Society of Americaformat or Malacara format or other available Zernike polynomialexpansion formats. Apart from the Zernike method of constructing thewavefront phase, any other non-Zernike method of wavefront phaseconstruction can also be adopted i.e. Fourier expansion, Taylorexpansion, etc.

(F) Global RIQ Metric Integrated Myopic Impetus Exposure Time

All the factors included in the above RIQ variants include wavefrontaberration, chromaticity and spectral sensitivity, Stiles-CrawfordEffect of the first kind, and optical/visual performance in theperipheral retina. Another factor that can be included is the amount oftime spent at various accommodative states on an average day (the dailyamount of near work), also known as the myopic impetus exposure time, T(A). This provides the following GRIQ variant:

$\begin{matrix}{\int_{Amin}^{Amax}{{T(A)}*{GRIQ}({dA})}} & {{Equation}6}\end{matrix}$

(G) Other Possible RIQ Measures

As mentioned above, other measures of RIQ proposed can also be used inthe design of devices, lenses and methods. One example of an alternativeRIQ measure is simple modulation transfer function (MTF). Referring toEquation 2, a polychromatic MTF is formed by computing the modulus ofreal part of the optical transfer function and in addition excluding thestep of convolution with the CSF function. A monochromatic MTF is formedif S(λ) is also removed from Equation 2.

2. Through Focus RIQ

RIQ may also be considered anterior and/or posterior to the retina. TheRIQ anterior and/or posterior to the retina is called ‘through focusRIQ’ herein. Similarly, RIQ at and/or around the retina may also beconsidered over a range of focal lengths (i.e. when the eyeaccommodates, which causes refractive characteristics of the eye inaddition to the focal length to change).

Embodiments consider not only RIQ at the retina, but also the change inthrough focus RIQ. This is in contrast to an approach that may, forexample, consider only the RIQ at the retina and/or an integral orsummation of RIQ measures at or around the retina. For example,embodiments of the lenses, devices and methods described herein effect,or are designed to effect for an eye with particular refractivecharacteristics, a change in or control over the extent or rate ofchange in RIQ in the directions anterior to the retina (i.e. thedirection from the retina towards the cornea) and/or posterior to theretina.

Embodiments also effect, or are designed to effect a change in orcontrol over the variation in RIQ with focal distance. For exampleseveral candidate lens designs may be identified through effecting achange in the retinal image quality in the direction posterior to theretina and then a single design or subset of designs may be identifiedtaking account of variation in RIQ with change in focal length.

In other embodiments the process described above is reversed. Inparticular, a set of designs is selected based on changes in RIQ at theretina with focal distance. Selection within the set is then made withreference to the through focus RIQ.

In still other embodiments a single evaluation process is conducted thatcombines consideration of through focus RIQ and changes of RIQ at theretina with a said focal distance. For example, an average measure ofRIQ with changes in focal distance may be used to identify a design. Theaverage measure may give more weight to particular focal distances (e.g.distance vision, intermediate vision and near vision and therefore maybe weighted differently),

In various embodiments, through focus and/or changes of RIQ at theretina with focal distance are considered: i) on-axis, ii) integratedaround on-axis, for example in an area corresponding to or approximatinga pupil size, with or without consideration of the Stiles-Crawfordeffect, iii) off-axis (where off-axis means a location, set of locationsor integral of locations on the retina outside the fovea, which may bewhere light at field angles more than about 10 degrees is focused),and/or iv) for a combination of i) to iii).

While the description herein refers predominantly to quantitativemeasures of RIQ, qualitative measures may be used in the design processof an aberration profile instead or in addition to the quantitativemeasures. For example, the Visual Strehl Ratio at a particular throughfocus location is computed based on the point spread function. As can beseen from the example images referred to in the following section, thepoint spread function can be visually evaluated. This provides for amethod of qualitatively evaluating through focus.

3. Aberrations Affecting Image Quality at the Retina and Through FocusRIQ

The influence of lower order aberrations on retinal image quality andthrough focus RIQ is well understood. The use of corrective lower orderaberrations represents a traditional method of refractive errorcorrection for an eye. Accordingly, the identification of an aberrationprofile consisting of lower order aberrations to correct for defocus andastigmatism will not be described herein in detail.

The influence of higher order aberrations (HOA) on image quality can beappreciated from the through-focus two-dimensional point spreadfunctions 300 illustrated in FIG. 3 . In FIG. 3 the rows show the pointspread functions for a selection of aberrations and the horizontal axisshows the extent of defocus for the relevant aberration, in Diopters.

The point spread functions without any higher order aberrations 302 (inthe illustrated example images at the retina in an eye with myopia orhyperopia alone), with vertical coma 306 alone, and with horizontaltrefoil 308 alone, remain symmetrical with positive and negativedefocus. With positive and negative primary spherical aberrations,either alone 304 or in combination 310 with coma and/or trefoil, thethrough-focus in the point spread function is asymmetrical for positiveand negative defocus. With certain HOA positive and negative defocus hasunequal effects on the image quality. It can be seen that these unequaleffects are more pronounced for spherical aberrations. The HOA thatexhibit asymmetrical effects on RIQ, visual acuity and/or contrastsensitivity have particular application to the lenses, devices andmethods disclosed herein.

The interactions occurring between HOA and defocus influence thethrough-focus RIQ. Some HOA interact favourably with defocus to improveRIQ, while others interact unfavourably to cause RIQ degradation. Themost commonly measured higher order ocular aberrations include sphericalaberration, coma and trefoil. Apart from these, the HOA profilesobtained with some multifocal optical designs precipitate considerablemagnitudes of wavefront aberrations, often expressed up to the 10thorder in Zernike polynomial representation.

Generally, in the Zernike pyramid, the terms closer to the centre areoften more influential when gauged in terms of the resultant opticaleffects than those at the edge/corner. This may be because the termsfarther away from the centre have a relatively large planar area on thewavefront compared to those whose angular frequency is closer to zero.Accordingly, Zernike terms that have the highest potential to interactwith defocus are, for example, the terms with even radial order havingzero angular frequency component, i.e. the fourth, sixth, eighth, andtenth order Zernike coefficients, representing primary, secondary,tertiary and quaternary, spherical aberrations.

The foregoing description of aberrations identifies some of theaberrations that affect retinal RIQ and through focus RIQ. Thedescription is not, nor is it intended to be, an exhaustive descriptionof all aberrations that affect retinal RIQ and through focus RIQ. Invarious embodiments additional aberrations that affect the retinal RIQand/or through focus RIQ may be considered, the relevant aberrationsbeing identified having regard to the current refractive state of theocular system (meaning the eye together with any lenses or opticaldevices that affect the wavefront received by the retina) and a targetretinal RIQ/through focus RIQ.

4. Optimising RIQ

When designing or selecting a required change in refractive state of aneye a measure of RIQ and through focus RIQ is required. In particular,finding a magnitude and sign of defocus of the relevant aberrations thatproduces an acceptable RIQ and through focus RIQ is required. The searchis for the best or at least an acceptable combination of RIQ and throughfocus RIQ. In some embodiments described herein, a merit functionS=1/RIQ is used for this purpose.

Identifying aberration coefficients that optimise RIQ at the retina maybe achieved by finding a minimum value of the function S. Consideringthis in combination with through focus RIQ adds complexity to theoptimisation process. Various methods can be used to address thiscomplexity.

One example is to use a non-linear, unconstrained optimization routine,over the chosen group of Zernike SA coefficients as variables. A randomelement, either automatic or through human intervention may beincorporated to shift to different locations so as to find alternativelocal minima of the function S. The criteria by which the optimisationroutine evaluates performance may be a combination of retinal RIQ andkeeping the through focus RIQ within predefined bounds of the retinalRIQ. The bounds may be defined in various ways, for example as a rangeabout the value for retinal RIQ. The range may be fixed (e.g. plus orminus 0.15 for Visual Strehl ratio or GRIQ or similar measure), or mayvary (e.g. be within a defined rate of change with increasing distancefrom the retina). As explained in more detail herein below, theobjective range for through focus RIQ may change depending on whetherthe objective is to provide a sloped through focus RIQ so as to providestimulus to inhibit or encourage eye growth under an optical feedbackexplanation of emmetropisation, or to provide a flat through focus RIQ.

Another approach is to limit the number of possible aberration profiles.One way of limiting the possible aberration values is to specify thatthe Zernike coefficients can only have values corresponding toincrements of 0.05 μm focus, or another increment interval. The intervalcan be selected having regard to the available computational resources.By limiting the number of allowable coefficient values it is possible tosimulate the performance of all aberration profiles formed by thecombinations of Zernike coefficients, following which those with thebest or acceptable on-axis RIQ and through focus RIQ can be identified.The results of this process may be used to constrain more fine-tunedanalysis, for example by returning to an optimisation routine withcoefficient values within a small range around an identified candidatecombination.

5. Controlling Stimulus for Refractive Error Progression by OpticalFeedback

In some embodiments a lens, device or method that incorporates anaberration profile to provide a particular on-axis RIQ and through focusRIQ is applied to an eye with progressing myopia or an eye identified asat risk of developing myopia. A person may be identified as being atrisk of developing myopia based on a number of indicators, includingwhether their parents experienced myopia/progressing myopia, theirethnicity, lifestyle factors or otherwise. A person may be identified asbeing at risk of developing myopia if their eye(s) have an RIQ thatimproves in the direction of eye growth (when eye growth is notrequired), either with or without any correction that is currently beingused (e.g. with or without a current prescription of lens). The use ofimproving RIQ in the direction of eye growth may be used alone or inconjunction with other indicators, for example the other indicatorslisted above.

Under an optical feedback mechanism explanation of emmetropisation basedon RIQ, the eye is stimulated to grow to the position where the meritfunction S is minimised. Under this explanation of emmetropisation, forhuman eyes, if the location of the minimum of the function S (which maybe a local minimum) is posterior to the retina or if through focus RIQimproves posterior to the retina, the eye will be stimulated to growlonger and if this location is on or anterior to the retina, then theeye will remain at the same length. The lenses, devices and methodsdisclosed herein may be applied to provide stimulus under this opticalfeedback mechanism explanation of emmetropisation. Embodiments foraddressing eye growth under the optical feedback explanation ofemmetropisation (e.g. to address myopia progression or to seek tostimulate eye growth to correct hyperopia) use aberrations to affect oneor both of the location of the minima of the function S relative to theretina and the gradient of the function S through the retina.

The following description describes how combinations of selected HOA canaffect a change in a measure of through focus RIQ. Each of theseaberrations can readily be incorporated into a lens, optical device orused in a method of changing the aberration profile of light received bythe retina. This provides a mechanism by which a lens, optical device ormethod of changing the refractive state of an eye (e.g. refractivesurgery) can be designed or selected to obtain a target through focusRIQ for an eye, or at least change the refractive state of the eye tocloser to a target through focus RIQ. As described in more detail below,achieving a target through focus RIQ is considered together withachieving or getting closer to a target on-axis RIQ at the retina forparticular focal length, which is typically distance vision (objects atinfinity or practically for human eyes, greater than 3 meters to about 6meters), but which may be another selected focal length, for exampleintermediate vision (e.g. about 0.75-2 meters) or near vision (e.g.about 0.35 to 0.60 meters).

For the examples described below the RIQ was evaluated using the VisualStrehl Ratio shown in Equation 2.

(A) Primary Spherical Aberration, Coma and Trefoil

The interactions between primary spherical aberration, coma and trefoiland their affect on eye growth can be described using a wavefront phasefunction defined using defocus, primary spherical aberration (PSA), comaand trefoil terms of a standard Zernike expansion.

The pupil size was fixed at 4 mm and the calculations were performed at589 nm wavelength. For the purposes of evaluating affects of aberrationprofiles on ocular growth, it was assumed that a location of a minimumof the above described function S posterior to the retina provides astimulus to grow to that location and that there will not be stimulusfor eye growth if the minimum of the function S is on or in front of theretina. In other words, it is assumed that the image formed on theretina provides a stimulus to grow to minimise the function S. The rangeof values of PSA, horizontal and vertical coma, and horizontal andvertical trefoil that were used in the simulations are:

PSA=(−0.30, −0.15, 0.00, 0.15, 0.30) μm

Horizontal Coma=(−0.30, −0.15, 0.00, 0.15, 0.30) μm

Vertical Coma=(−0.30, −0.15, 0.00, 0.15, 0.30) μm

Horizontal Trefoil=(−0.30, −0.15, 0.00, 0.15, 0.30) μm and

Vertical Trefoil=(−0.30, −0.15, 0.00, 0.15, 0.30) μm.

With a total of 3125 combinations tested, overall it was observed thatspherical aberration primarily governed the direction of improving RIQ.

FIGS. 4 to 7 illustrate the stimulus for eye growth resulting fromthrough-focus RIQ for a selection of the combinations, in particular thecombined effects of PSA together with horizontal and vertical coma, andtogether with horizontal and vertical trefoil. FIGS. 4 to 7 are on acontinuous scale and white (0) indicates no progression andgray-to-black transition indicates the amount of progression inDiopters.

FIG. 4 shows a graph 400 of the interaction of primary sphericalaberration and horizontal coma. The gray plot indicates the amount ofprogression of myopia that is stimulated by the combination of these twoaberrations, where white 402 indicates no stimulus for progression andshades towards black 404 indicate stimulus for progression of myopia (inthis case up to −0.8 D) as a result of PSA combined with horizontalcoma. FIG. 5 shows a graph 500 of myopia progression as a function ofthe interaction of primary spherical aberration and vertical coma. Likein FIG. 4 , white areas 502 indicate no stimulus for progression anddark areas 504 indicate stimulus for progression. FIG. 6 shows a graph600 of the interaction of primary spherical aberration and horizontaltrefoil. FIG. 7 shows a graph 700 of myopia progression as a function ofthe interaction of primary spherical aberration and vertical trefoil.For the combinations shown in FIGS. 4 to 7 , about 52% of thecombinations provide stimulus to encourage eye growth.

The above described stimulus for eye growth may accordingly be removedby controlling the refractive state of an eye to be within any of thewhite areas in FIGS. 4 to 7 . This may be achieved, for example, bydesigning a lens or optical device that when applied modifies therefractive characteristics of the eye, to result in the retina of theeye experiencing a through focus RIQ that does not improve in thedirection of eye growth (posterior to the retina) or which decreases inthe direction of eye growth.

Although trefoil and coma in the range of −0.30 to 0.30 μm over a 4 mmpupil do not appear to have a significant impact on the direction ofgrowth (the maximum progression effect is only −0.1D), positive PSAseems to accelerate growth while negative PSA inhibits growth. The PSAtherefore appears to have the dominant effect. Accordingly, at least foran eye with positive PSA and optionally one of coma and trefoil, addingnegative PSA will inhibit eye growth under the optical feedbackexplanation of emmetropisation. It follows that providing negative PSAto an eye, or at least removing positive PSA may remove the stimulus foreye growth. Any coma and trefoil in the eye may be left unchanged oroptionally partially or fully corrected (preferably within the range of−0.30 to 0.30 μm).

(B) Spherical Aberration and Astigmatism

To illustrate the interactions between primary spherical aberration andastigmatism, a wavefront phase function was defined using theseaberrations (including both horizontal/vertical and oblique components)and defocus. FIGS. 8 to 13 (unlike FIGS. 4 to 7 ) are on a binaryscale—where white (1) indicates test cases that cause stimulus forprogression (i.e. increase in ocular growth) and black (0) indicatescandidate combinations that result in no progression (i.e. no oculargrowth stimulus or a stop signal). The scale has no units.

FIG. 8 shows a graph 800 indicating the magnitude of myopia progressionfor PSA vs. a primary oblique astigmatic component (POA) vs. a primaryhorizontal/vertical astigmatic (PHV) component. The graph 800 indicatesthose combinations of PSA and astigmatism that will result in stimulusfor myopia progression (white) and those combinations that will notresult in stimulus for myopia progression (black). Neither POA nor PHVappear to have a significant impact on the effects of PSA.

FIG. 9 shows a graph 900 indicating the magnitude of myopia progressionfor PSA vs. a secondary oblique astigmatic (SOA) component vs. asecondary horizontal/vertical astigmatic (SHV) component. Neither SOAnor SHV appear to have a significant impact on the effects of PSA.

A stimulus for eye growth may accordingly be removed by controlling therefractive state of an eye to be within any of the white areas in FIGS.8 and 9 .

From FIGS. 8 and 9 , the primary and secondary astigmatic componentshave a small influence on enhancing or inhibiting eye growth, whencombined with PSA. Accordingly, considering these aberrations, thisindicates priority should be provided to PSA. In addition, it may bedetermined whether the eye has high levels of POA, PHV, SOA and/or SHV.If this is the case, then correcting these aberrations (by reducing orsubstantially eliminating them) may also assist in removing stimulus foreye growth.

(C) Higher Order Spherical Aberrations

For unaided or single-vision spectacle corrected eyes a fourth orderZernike expansion is generally sufficient to describe the wavefront atthe exit pupil. However, this is not necessarily the case when contactlenses are used for correction, especially with multifocal contactlenses (both aspheric and concentric), substantial amounts of fifthorder and higher HOA may be imposed. Multifocal contact lenses may forexample be described using up to about the tenth or twentieth order ofZernike polynomials. In such cases the magnitudes and signs of thehigher order spherical aberrations start to play a significant role (inaddition to PSA).

To illustrate the interactions between primary, secondary, tertiary andquaternary spherical aberrations of a standard Zernike expansion, awavefront phase was defined using these terms and defocus. Severalcombinations of HOA as predicted from modelled data with such multifocalcontact lenses were used. Selective sets of these HOA that demonstrateinteractions to produce peak RIQ were obtained via dedicated non-linearoptimization routines. All the calculations were performed over a 4 mmpupil, and at 589 nm wavelength. It was observed that at least the firstthree modes of spherical aberration of the inherent eye played asignificant role in governing the direction of stimulus for eye growthand in some cases higher modes of spherical aberration also played asignificant role.

The results described below relate to secondary spherical aberration(SSA), tertiary spherical aberration (TSA) and quaternary sphericalaberration (QSA), but spherical aberrations with higher orders may alsobe used in embodiments of the lenses, devices and methods describedherein. For all four types of spherical aberrations, a range from −0.30to 0.30 μm was used to investigate the effects of the combinations ofHOA. These ranges for these types of aberrations do not necessarilyaccord with normative distributions of aberrations associated with eyesbecause the occurrence of these higher order aberrations are notnecessarily associated with the eyes but with the optical devices (suchas multifocal contact lenses) alone or in combination with the eyes.Furthermore, the range from −0.30 to 0.30 μm is merely used toillustrate the effects, but when determining combinations of HOA toprovide an aberration profile in a lens or optical device, or to beeffected by surgical procedures, larger or smaller ranges may be used.

FIGS. 10 to 12 show the stimulus for myopia progression as a function ofPSA together with SSA, TSA and QSA respectively. This schema is a binarycolour plot, where white (0) indicates wavefront aberration combinationsthat provide stimulus for myopia progression under the feedbackmechanism described above and black (1) indicates combinations thatdiscourage myopia progression. From these graphs it is apparent that thehigher orders of spherical aberrations have an impact on the stimulusfor progression of myopia. About 82% of the combinations investigatedsuggest stimulus for eye growth. Interactions of the sphericalaberration terms depend on their individual signs and then theirindividual magnitudes.

FIG. 10 shows a graph 1000 indicating the presence of stimulus formyopia progression as a function of combinations of PSA and SSA. In FIG.10 , it can be seen that when PSA in the range −0.30 μm to 0.20 μm iscombined with negative SSA ranging from 0.00 to −0.30 μm, there is noimprovement of RIQ in the direction of eye growth, thus no myopiaprogression is predicted (i.e., in the area indicated 1004). However,when PSA ranging from 0.20 to 0.30 μm is considered with negative SSA ofabout −0.10 μm, it seems to aggravate the progression, as indicated inthe area 1002. Overall, the sign of SSA seems to have a governing effecton the effect of the wavefront aberrations and the resultant retinalimage quality. Negative SSA of considerable magnitudes (i.e. greaterthan −0.20 μm) predicts a protective effect against myopia progressionwhen combined with either positive or negative PSA, when PSA and SSA arethe only two HOA involved in the wavefront aberration of the candidateeye.

FIG. 11 shows a graph 1100 indicating the presence of stimulus formyopia progression as a function of combinations of PSA and TSA. WhenPSA and TSA have the same sign and TSA is about ⅘th of PSA in magnitude,as indicated by rectangular box 1106, no or little myopia progression ispredicted (black area). However, with other combinations of PSA and TSA,for example as indicated in areas 1102 and 1104, myopia progression canbe expected.

FIG. 12 shows a graph 1200 indicating the presence of stimulus formyopia progression as a function of combinations of PSA and QSA. WhenPSA and QSA have opposite signs and QSA is about ⅘th of PSA inmagnitude, as indicated by the predominantly black area 1204, no myopiaprogression is predicted. However, with other combinations of PSA andQSA, (for example as indicated in white areas 1202 and 1206) myopiaprogression can be expected.

FIG. 13 is a graph (1300) showing the presence of stimulus forprogression of myopia as a function of PSA, SSA and TSA. This schema isa binary colour plot, where 1 (white) indicates wavefront aberrationcombinations that favour myopia progression; while 0 (black) indicatescombinations that discourage myopia progression (i.e. do not providestimulus for eye growth).

The majority of the black filled circles 1304 are in the region governedby negative SSA, with a few exceptions. Further, combinations in whichPSA and TSA have the same sign coupled with negative SSA seem to providea protective effect against myopia progression. The combinations of PSA,SSA, TSA and QSA that have a protective effect against myopiaprogression under the optical feedback explanation of emmetropisation(which include the black areas shown in FIG. 13 ) can be summarised asshown in the Table 1.

TABLE 1 Combination sets of higher order aberrations which discouragethe eye growth (i.e potential treatment for myopia). Specific higherorder aberration in Magnitude and sign of the SNo addition to defocushigher order aberration 1 PSA only −0.30 μm <= PSA < 0.125 μm 2 SSA only−0.30 μm <= SSA <= 0.075 μm 3 TSA only −0.30 μm <= TSA <= 0.075 μm 4 QSAonly −0.10 μm <= QSA <= 0.075 μm 5 PSA & SSA −0.30 μm <= PSA <= 0.20 μmand −0.25 μm <= SSA <= 0.025 μm 6 PSA & TSA −0.30 μm <= PSA <= 0.30 μmand TSA = (PSA/2) μm +/− 0.075 μm 7 PSA & QSA −0.30 μm <= PSA <= 0.30 μmand QSA = (|PSA|/3) μm +/− 0.075 μm 8 PSA, SSA, TSA −0.30 μm <= PSA <−0.05 μm & 0.05 μm < PSA < 0.30 μm; −0.30 μm <= SSA < 0.05 μm; −0.20 μm<= TSA < −0.025 μm & 0.025 μm < TSA < 0.20 μm; 9 PSA, SSA, TSA and −0.30μm <= PSA < −0.05 μm & QSA 0.05 μm < PSA < 0.30 μm; −0.30 μm <= SSA <0.05 μm; −0.20 μm <= TSA < −0.025 μm & 0.025 μm < TSA < 0.20 μm; −0.20μm <= QSA < −0.025 μm & 0.025 μm < QSA < 0.20 μm;

The majority of the white circles 1302 are in the region governed bypositive SSA, with a few exceptions. Further, combinations in which thePSA and TSA have the same sign coupled with positive SSA may provide atreatment effect for hyperopia. The combinations of PSA, SSA, TSA andQSA that have a treatment effect against hyperopia under the opticalfeedback explanation of emmetropisation (including the white areas shownin FIG. 13 ) can be summarised as shown in the Table 2.

TABLE 2 Combination sets of higher order aberrations which encourage eyegrowth (i.e potential treatment for hyperopia). Higher order aberrationin addition Magnitude and sign of the SNo to defocus higher orderaberration 1 PSA only 0.30 μm => PSA >= 0.125 μm 2 SSA only 0.30 μm =>SSA > 0.075 μm 3 TSA only 0.30 μm => TSA > 0.075 μm 4 QSA only −0.30 μm<= QSA <= −0.125 μm or 0.30 μm => QSA > 0.075 μm 5 PSA & SSA −0.30 μm <=PSA <= 0.30 μm and 0.30 μm >= SSA > 0.075 μm 6 PSA & TSA −0.30 μm <= PSA<= 0.30 μm and (PSA/2) μm + 0.075 μm <= TSA < 0.30 μm or −0.30 μm <= TSA< (PSA/2) μm − 0.075 μm 7 PSA & QSA −0.30 μm <= PSA <= 0.30 μm and QSAin the range −0.20 to 0.20 μm but excluding values where QSA = (|PSA|/3)μm +/− 0.075 μm 8 PSA, SSA, TSA −0.30 μm <= PSA < −0.05 μm & 0.05 μm <PSA < 0.30 μm; 0.075 μm <= SSA < 0.30 μm; −0.20 μm <= TSA < −0.025 μm &0.025 μm < TSA < 0.20 μm; 9 PSA, SSA, TSA −0.30 μm <= PSA < −0.05 μm &and QSA 0.05 μm < PSA < 0.30 μm; 0.075 μm <= SSA < 0.30 μm; −0.20 μm <=TSA < −0.025 μm & 0.025 μm < TSA < 0.20 μm; −0.20 μm <= QSA < −0.025 μm& 0.025 μm < QSA < 0.20 μm;

Accordingly, when designing a lens, optical device or method of alteringthe eye, the aberrations may be selected to provide a combination of theaforementioned aberrations that provide for either a protective effectagainst eye growth, or which encourage eye growth. The combination ofaberrations may be applied in combination with the required correctionof any myopic defocus or hyperopic defocus.

From the foregoing description, it is apparent that the sphericalaberration terms, including the primary, secondary, tertiary andquaternary SA terms influence RIQ and through focus RIQ. In addition, ithas been found that much higher orders of spherical aberration alsoinfluence RIQ and through focus RIQ. Accordingly in various embodimentsdifferent combinations of spherical aberration are used, includingembodiments using any combination of two or more spherical aberrationterms that provide a required or acceptable through focus RIQ profile,together with a required or acceptable RIQ at a particular focal length(e.g. distance vision).

6. The Instantaneous Gradient of the Image Quality

The foregoing description of stimulus for eye growth under the opticalfeedback mechanism explanation of emmetropisation focused on thelocation of the peak RIQ. Another approach is to consider the slope ofthrough-focus RIQ at the retina. In some embodiments, methods anddevices control or utilise this gradient of the image quality metric.The gradient may be considered for any measure of RIQ.

In the following description it is assumed that a positive measure ofthe gradient of the through-focus RIQ (increasing RIQ posterior to theretina) provides a stimulus for the development and progression ofmyopia, while a negative measure of the same retards or halts myopiaprogression.

FIG. 14 shows a plot of RIQ for two different cases, 1402 and 1404, as afunction of through focus in the direction posterior to the retina. Thecases are two different combinations of PSA, SSA and TSA that produceidentical retinal RIQ. As can be seen from the figure, although bothsets of selected aberrations produce similar image quality at the retina(defocus=0), with the introduction of defocus (in the direction of eyegrowth) the retinal image quality of test case 1402 ramps up indicatingstimulus for eye growth, while test case 1404 indicates that there wouldbe no stimulus for growth, as the retinal image quality degrades furtherin the direction of eye growth, i.e. positive Zernike defocus.

From the results described above that indicate the effects of HOA onimage quality and the resulting progression of myopia, it is possible todetermine the relevant HOA combinations that can be used in lenses,optical devices, or effected using optical surgery, which, whererelevant in combination with the eye's aberrations, will result in theHOA combinations that inhibit or retard eye growth for the treatment ofmyopia progression. In order to slow down eye growth in myopia,compensating optical devices or surgical procedures can be used that, incombination with the optics of the eye, will result in a combination ofHOA that results in a negative gradient of through-focus retinal imagequality, as shown in example 1404 (FIG. 14 ). For treating hyperopia,compensating optical devices or surgical procedures can be used that, incombination with the optics of the eye, will result in a combination ofHOA that results in a positive gradient of through-focus retinal imagequality, as shown in example 1402 (FIG. 14 ).

If an aberration profile has a varying RIQ across a through focus range,then the slope of through focus RIQ at a particular focal length can bechanged by selecting a suitable defocus term C(2,0) with the consideredRIQ profile. For example, if the slope is positive at a first level ofthrough focus and negative at a second level of through focus, the slopeat the retina of a recipient eye can be selected by selectivelyintroducing defocus at either the first or second level. Examples ofaberration profiles that have varying RIQ slopes at different levels ofdefocus are provided below in relation to embodiments of aberrationprofiles for application to presbyopia. Many of the embodimentsdescribed for presbyopia may be applied to provide a stimulus to retardor encourage eye growth under the optical feedback explanation ofemmetropisation described above. Typically, younger people haveprogressing myopia and as such they will not be experiencing presbyopia.Accordingly, the aberration profile selected may place less weight onachieving high RIQ over a large through focus range and more weight onachieving the highest RIQ at the retina for distance vision incombination with providing a negative slope RIQ profile through theretina (i.e. decreasing RIQ in the direction of eye growth). For theyoung hypermetropes, again, the selected aberration profile may placeless weight on achieving high RIQ over a large through focus range andmore weight on achieving the highest RIQ at the retina for distance incombination with provision of a positive slope of RIQ profile behind theretina (in the direction of eye growth).

In addition, the slope across a range of field angles can be consideredand/or variations in the RIQ for a range of pupil sizes. For example, anaberration profile may be selected that provides an average, mode orsubstantially uniform slope across a range of field angles, such as 10,20, 30 or 40 degrees that either inhibits or encourages eye growth(and/or cancel existing aberrations in the eye that encourage or inhibiteye growth respectively). The average slope across the range of pupilsizes or at the mode pupil size may also be considered. Alternatively,the design may be selected that has either a positive or negative slopeof through focus RIQ for all field angles within a range and/or for allpupil sizes with a range.

7. Aberration Design or Selection Process

In some embodiments determining the aberration profile required in alens, optical device or resulting from a procedure, includes firstidentifying the HOA present in the eye. Measurements may be taken, forexample, using wavefront eye exams that use aberrometry such as with aShack-Hartmann aberrometer. The eye's existing HOA may then be takeninto account. In addition, any HOA effects inherent in the lenses oroptical devices may also be taken into account.

When the requirement is for a lens that provides stimulus for eye growthor to retard eye growth, these existing HOA are then compared to HOAcombinations that inhibit or retard myopia progression (for example asdiscussed above with reference to FIGS. 5 to 14 ) to determine one ormore additional HOA that may be required to reduce or retard orencourage eye growth under the optical feedback mechanism ofemmetropisation. These additional combinations are then implemented inthe design of lenses or optical devices or implemented using opticalsurgery. Flowcharts in FIGS. 15 and 16 provide a summary of suitablemethods.

Alternatively, the eye's existing aberrations may be disregarded and anaberration profile that provides the required through focus RIQ slopemay be provided for the eye by a lens, preferably a removable lens sothat different aberration profiles may be trialled if required. Theaberration profile resulting from the combination of the aberrationprofile of the lens and the eye may then be measured to determine if theRIQ characteristics are acceptable (for example, provide a particularthrough focus RIQ slope and acceptable RIQ for distance vision).Alternatively, different lenses may be placed on the eye with measuresof objective and/or subjective vision determining which lens to select.Where the lens is selected to provide stimulus inhibiting or encouragingeye growth without regard to the eye's existing aberrations, theselected aberration profile may be one with generally higher values ofspherical aberration, so that the sign of the slope is not changed bylower level of HOA in the eye.

In other applications, the goal for the combination of HOA may bedifferent. For example, when considering presbyopia the goal may be acombination of aberrations that provide high RIQ over a large throughfocus. Where peripheral vision is important, then the objective mayinclude high RIQ over a large range of field angles. Accordingly, invarious embodiments the HOAs are utilised to optimise for the goals of acombination of high RIQ at the retina and one or more of a low slopethrough focus RIQ, a low change in RIQ with pupil diameter and a highRIQ in the peripheral field.

The examples that follow have been selected using the RIQ measure inEquation 2. The initial set of designs for analysis was found bycomputing this RIQ for all combinations of SA Zernike coefficients up tothe 10th order. Each coefficient was constrained to the range −0.3 μm to0.3 μm and constrained to be a value that is a multiple of 0.025 μm.

An analysis of the initial set of designs included: 1) identifyingoptimised combinations of Zernike coefficients that provide a high RIQand a negative slope through focus RIQ about the retina; 2)consideration of the RIQ and through focus RIQ and change in RIQ andthrough focus RIQ at different pupil sizes; and 3) consideration of theRIQ across the horizontal visual field. The relative weight given tothese stages of evaluation may vary for the particular recipient. Forthe purposes of identifying the following examples, most weight wasgiven to the first criteria.

8. Examples of Optical Designs Addressing the Slope of Through Focus RIQ

Examples of designs for affecting stimulus for eye growth under anoptical feedback mechanism are provided herein below. The examples beloware rotationally symmetric. However, astigmatic designs and othernon-rotationally symmetric designs may be produced. When a deliberatedecentration of the symmetric designs is imposed so that the opticalaxes of the correcting contact lens coincides with a reference axis ofthe eye say pupillary axis or visual axis, some residual amounts ofasymmetric aberrations like coma and trefoil can be induced, these maybe compensated by the choice of additional higher order asymmetricterms.

FIGS. 17 to 25 show the power profile graphs of sample designs thatprovide a RIQ that degrades in the direction of eye growth for on-axisvision (i.e. at zero field angle), thus providing a stimulus to inhibiteye growth under the optical feedback mechanism explanation of theemmetropisation process. The aberration profile graphs are described asthe axial power variation in diopters across the optic zone diameter.All the examples provided may have application to a progressing myopewhose spherical refractive error is −2.00D and this information isindicated by a dual gray line on all the power profiles.

FIG. 26 shows the details of a sample design that could be used forhyperopia treatment. This designs was produced by taking a specificaberration profile as an input parameter that would produce a positivegradient of through-focus retinal image quality in the direction of eyegrowth, as indicated in Table 2 and optimising the power profile (frontsurface of correcting contact lens) to achieve a required positivegradient. The lens design is described as the axial power variation indiopters across the optic zone diameter. The example provided may haveapplication to a non-progressing hyperope whose spherical refractiveerror is +2.00D and this information is indicated by a dual gray line onthe power profile.

As explained above, the example power profiles shown in FIGS. 17 to 26were selected based on the slope of RIQ around the retina. Across theseexamples, substantial variations in the value of RIQ can occur. Thesevariations occur on-axis, across the pupil diameter, and at differentfield angles. Additional selection criteria are the value of RIQ and thechange in RIQ with field angle. In particular the selection may be madeto maximise one or more of RIQ on-axis, across the pupil diameter (withor without reduction in light of the Stiles-Crawford effect) and atdifferent field angles. In addition, the size of the pupil of therecipient may also be used as a selection criterion—e.g. a firstaberration profile may better suit a first recipient with a normal pupilsize of 4 mm and a second aberration profile may better suit a secondrecipient with a normal pupil size of 5 mm. The ‘normal’ pupil size mayoptionally be selected having regard to lifestyle factors, such as theamount of time a person spends indoors versus outdoors. Additionalexamples referred to below incorporate these selection criteria. Firsthowever, to provide a point of comparison, the RIQ performance of asingle vision lens is described and shown in FIG. 27 .

FIG. 27 shows a graph of a measure of a through focus RIQ metric, whichin this case, and in each of the following examples, is Visual StrehlRatio (monochromatic). The RIQ may result, for example, from a singlevision contact lens with a power of −2.00 D used to correct a recipientmodel myopic eye with −2.00 D only. The horizontal (independent) axisshows the through focus, in Diopters. The zero (‘0’) value on thehorizontal axis represents the location of the focal point of the singlevision lens and the vertical (dependent) axis shows the RIQ. Three plotsare provided, one for on-axis (circles), one for a field angle of 10degrees (triangles) and one for a field angle of 20 degrees (crosses).Herein, the term ‘global’ is used to refer to consideration across arange of field angles, including zero. Thus, the graph shows ‘Globalthrough focus RIQ’, as it includes plots across a range of field angles.While a single vision lens has symmetrical RIQ on-axis at zero fieldangle, it has asymmetrical through focus RIQ at non-zero field angles,including both at 10 and 20 degrees. In particular, the graph shows thatRIQ improves in the direction of eye growth at non-zero field angles.Under the optical feedback mechanism explanation of emmetropisation,such as those described in U.S. Pat. Nos. 7,025,460 and 7,503,655 (Smithet al), peripheral as well as on-axis vision provides a stimulus for eyegrowth.

FIG. 28 shows a graph of RIQ for an embodiment of a lens (named‘Iteration A1’) selected to address the optical feedback mechanismexplanation of emmetropisation where eye growth is to be discouraged(e.g. to address progressing myopia or to address a risk of developingmyopia). The data for FIG. 28 was prepared for a pupil size of 4 mm andto address the same level of myopia as for the Single Vision Iteration.Comparing FIG. 28 with FIG. 27 , the RIQ no longer improves in adirection of eye growth for non-zero field angles. In particular, theRIQ has a strong trend towards degrading in the direction of eye growthfor 10 degrees off-axis. While there may be a slight improvement or nosubstantially no change in RIQ about the retina at 20 degrees off-axis,the overall effect is strongly biased towards degrading RIQ in thedirection of eye growth. FIG. 29 shows a power profile that results inthe RIQ graph of FIG. 28 .

FIG. 30 shows a graph of RIQ for another embodiment of a lens (IterationA2) selected to address the optical feedback mechanism explanation ofemmetropisation. The data for FIG. 30 was prepared for a pupil size of 5mm.

FIGS. 31 and 32 show graphs of the RIQ for two other embodiment of alens (Iteration C1 and Iteration C2 respectively) selected to addressthe optical feedback mechanism explanation of emmetropisation, but inthis case to provide improving RIQ in the direction of eye growth (e.g.to provide a stimulus to an eye to grow to correct hyperopia). FIGS. 31and 32 show exemplary embodiments selected with different weights to theselection criteria. In the power profile that gives FIG. 31 , achievinga high on-axis RIQ was given more weight than achieving a high RIQacross a large range of field angles. In the power profile that givesFIG. 32 , more weight was given to providing a high RIQ across a largerange of field angles than to achieving a high RIQ on-axis.

Table 3 lists the defocus and higher order aberrations coefficients upto 20th order, in microns, over a 5 mm pupil diameter for the abovedescribed power profiles.

TABLE 3 Defocus and higher order Spherical aberration coefficients overa 5 mm pupil for a single vision lens and four exemplary embodimentsthat provide a required slope for through focus RIQ. Iteration C(2,0)C(4,0) C(6,0) C(8,0) C(10,0) C(12,0) C(14,0) C(16,0) C(18,0) C(20,0)Single Vision −1.800 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0000.000 Lens Iteration A1 −1.568 0.107 −0.017 −0.016 −0.022 −0.008 0.0260.005 −0.016 0.003 Iteration A2 −1.562 0.115 −0.011 −0.011 −0.019 −0.0070.025 0.004 −0.017 0.005 Iteration C1 1.468 −0.135 0.020 0.029 0.0360.011 −0.036 −0.008 0.022 −0.003 Iteration C2 1.468 −0.116 0.035 0.010−0.013 −0.030 −0.014 0.025 0.004 −0.016

9. Application to Presbyopia

Extending the through focus RIQ may provide particular benefit in thecontext of presbyopia. The reduced ability of the eye to accommodate maybe partially compensated/mitigated by using the extended through focusapproach described herein.

In some embodiments the through focus RIQ is extended further by takinga monocular approach. In particular, one eye may have aberrationsoptimised for distance vision and the other eye optimised for nearvision. This optimisation is achieved by selecting different base powers(i.e. effective refractive prescriptions) for the lenses. The extendedthrough focus of each lens allows the base powers to be separatedfurther without sacrificing intermediate vision between the two basepowers. Under the monocular approach, selection of an aberration profilemay give a higher priority to the consideration of the RIQ and throughfocus RIQ, and change in RIQ and through focus RIQ at different pupilsizes (which reflect the change in the eye with differentaccommodation).

Similarly, a lens or optical device may be designed as a bifocal ormultifocal lens, with one or both of the parts incorporating aberrationprofiles as described herein to extend through focus RIQ. A combinationof bifocal/multifocal lenses or devices and the monocular approach canincrease the range of vision. For example, with reference to bifocallenses one eye may have far distance vision in the upper quadrants andnear vision in the lower quadrants and the other eye may haveintermediate vision in the upper quadrants and near vision in the lowerquadrants. The two lower quadrants may optionally have different basepowers, for example set at 2.00 D and 1.00 D.

When different lenses or different parts of lenses are used together,the base powers may be selected so that the through focus RIQ overlaps.For example, the base powers may be selected so that in combination theVisual Strehl Ratio does not drop below 0.10, 0.20, 0.30, 0.40 oranother selected value, between the combined RIQ profiles.

A) Examples for Presbyopia

FIG. 33 shows a graph of through focus RIQ (in this case Visual StrehlRatio) for seven power profiles. In this figure the vertical axis (RIQ)is defined on a logarithmic scale. FIG. 33 was obtained for a 5 mm pupilsize and an eye with no myopia or hyperopia and no other higher orderaberrations. Each power profile can be adapted to a myopic or hyperopiceye by incorporating an appropriate correcting defocus term, which doesnot affect the higher order aberrations defining the power profiles usedfor form FIG. 33 .

The seven power profiles are: a power profile that may appear in aconventional centre-distance aspheric multifocal lens (indicated bytriangles in FIG. 33 ); a power profile that may appear in aconventional centre-near multifocal lens (indicated by ‘x’ in FIG. 33 );a power profile that may appear in a centre-distance concentric bifocallens (indicated by filled ‘□’ in FIG. 33 ); a power profile that mayappear in a centre-near concentric bifocal lens (indicated by empty‘< >’ in FIG. 33 ) and three iterations (Iteration B1, Iteration B2,Iteration B3) including a favourable combination of spherical aberration(indicated by filled circles, bold ‘+’ signs and a concentric circlepairs, respectively, in FIG. 33 ).

The power profiles for each of these are shown in FIGS. 34 to 40 . Thecentre-distance and centre-near aspheric multifocals had the centrecomponent extend to about 2 mm and the outer zone power commence at aradius of about 1.8 mm. A linear transition was provided between thenear and distance power zones. The concentric bifocals both had a ringstructure, alternating between an additional power of 2 Diopters and noaddition power.

Table 4 lists the defocus and higher order spherical aberrationcoefficients up to 20^(th) order, in microns, over a 5 mm pupildiameter, for the three exemplary embodiment power profiles, namely:Iteration B1 (FIG. 38 ), Iteration B2 (FIG. 39 ) and Iteration B3 (FIG.40 ), respectively.

TABLE 4 Defocus and Spherical aberration coefficients of three exemplaryembodiments for presbyopia Iteration C(2,0) C(4,0) C(6,0) C(8,0) C(10,0)C(12,0) C(14,0) C(16,0) C(18,0) C(20,0) Iteration B1 −0.096 −0.135 0.0200.029 0.036 0.012 −0.036 −0.010 0.022 0.000 Iteration B2 −0.092 0.0320.074 −0.015 −0.006 −0.018 −0.009 0.007 0.011 0.002 Iteration B3 0.0330.003 0.077 −0.045 −0.023 0.010 0.014 0.007 0.003 −0.014

Table 5 lists out the defocus and higher order spherical aberrationcoefficients up to 20^(th) order, in microns, over a 5 mm pupildiameter, for the described power profiles, namely, centre-distanceaspheric multifocal (FIG. 34 ), and centre-near aspheric multifocal(FIG. 35 ), respectively.

TABLE 5 Defocus and Higher order spherical aberration coefficients ofboth centre-distance and centre-near type aspheric multifocal lensesIteration C(2,0) C(4,0) C(6,0) C(8,0) C(10,0) C(12,0) C(14,0) C(16,0)C(18,0) C(20,0) Centre-Distance 1.150 0.181 −0.090 0.020 0.000 0.0000.000 0.000 0.000 0.000 bifocal Centre-Near 0.324 −0.244 0.114 −0.021−0.013 0.011 0.000 0.000 0.000 0.000 bifocal

In the aspheric multifocal lenses the spherical aberration coefficientsprogressively decrease in absolute magnitude with an increase in order.This is in contrast to the power profiles of Iteration B1, Iteration B2and Iteration B3, which include at least one higher order sphericalaberration term with an absolute value coefficient greater than theabsolute value of the coefficient for a lower order term. Thischaracteristic is present in many embodiments of power profile describedherein.

From FIG. 33 , it can be noted that the centre-distance asphericmultifocal has a RIQ of 0.23 for distance vision, which substantiallyinferior than the other power profiles. However, performance of thislens as gauged by the RIQ metric is maintained relatively constant overa large through focus range. For example, at −0.4 Diopters the RIQ isabout 0.2, at 0.67 the RIQ is about 0.18 and at −1.0 Diopters, the RIQis about 0.12.

The centre-near aspheric multifocal has a RIQ for distance vision, ofabout 0.50. The centre-near bifocal falls to an RIQ of about 0.24 at−0.67 Diopters (still better than the centre-distance asphericmultifocal). However, beyond that the centre-near aspheric multifocalhas a rapidly decreasing RIQ and at −1.0 Diopters has an RIQ of about0.08.

Both of the concentric bifocals (centre-distance and -near) have a lowRIQ of 0.13 and 0.21 for distance vision. Both of the concentricbifocals maintain their level of RIQ or better over a range ofapproximately 1.1 Diopters.

Iteration B1, Iteration B2 and Iteration B3 all have at least as goodRIQ at distance vision as the centre near bifocal and better RIQ as theeye accommodates. For example Iteration B2 has an RIQ of about 0.53 at−0.40 Diopters, about 0.32 at −0.67 Diopters and about 0.13 at −1.0Diopters. However, the through focus performance of each of IterationB1, Iteration B2 and Iteration B3 can be further extended. Thisextension is achieved by shifting the curves to the left in FIG. 33 .However, the performance of the centre-near aspheric multifocal lenscannot be shifted in this manner without substantially affectingperformance, due to the asymmetric RIQ that decreases substantially morerapidly for plus powers (right hand side of FIG. 33 ).

For example, all three of these iterations have an RIQ of about 0.40 at+0.55D. Combining the spherical aberration terms with a +0.55D defocusterm will shift the RIQ value for distance vision to the value for +0.55D in FIG. 33 . Taking again Iteration B2, the through focus performanceis modified as follows: an RIQ of about 0.40 at distance vision, an RIQof about 0.53 at −0.40 Diopters, about 0.64 at −0.67 Diopters, about0.52 at −1.0 Diopters, about 0.40 at −1.1 Diopters, and about 0.15 at−1.5 Diopters.

Accordingly, by shifting the distance vision point in a lens withcombinations of HOA that extend through focus RIQ performance, then thelenses, devices and methods that provide the combination of HOA can havea substantially improved through focus performance. This is achievedwhile maintaining at least as good RIQ as a centre near asphericmultifocal and substantially improved RIQ in comparison to a centredistance aspheric multifocal. The amount of defocus plus power added toshift the RIQ curves is a matter of choice, representing a trade-offbetween distance vision RIQ and near vision RIQ.

Table 6 shows the defocus (leftmost column) and RIQ values for each ofthe power profiles described above. It also shows the defocus valuesshifted by +0.55D, applicable when to Iteration B1, Iteration B2 and/orIteration B3 are modified by this amount.

TABLE 6 RIQ values for two bifocal lenses, two concentric bifocal lensesand three aberration profiles for extended through focus RIQ Centre-Centre- Centre- Centre- Distance Near Distance Near Defocus Defocusaspheric aspheric Iteration Iteration Iteration concentric concentricshifted (D) multifocal multifocal B1 B2 B3 bifocal bifocal by +0.50−1.1085 0.1021 0.0601 0.1342 0.0918 0.0971 0.2025 0.1349 −0.6085 −0.99770.1212 0.0768 0.1831 0.1338 0.1228 0.2447 0.1524 −0.4977 −0.8868 0.14070.1062 0.2394 0.1882 0.1577 0.2913 0.1675 −0.3868 −0.7760 0.1598 0.15740.2957 0.2511 0.2095 0.3362 0.1789 −0.2760 −0.6651 0.1776 0.2383 0.34230.3160 0.2830 0.3700 0.1851 −0.1651 −0.5543 0.1931 0.3481 0.3867 0.42620.3723 0.3839 0.1855 −0.0543 −0.4434 0.2060 0.4699 0.4550 0.5318 0.45830.3735 0.1805 0.0566 −0.3326 0.2162 0.5715 0.4992 0.6099 0.5266 0.34170.1709 0.1674 −0.2217 0.2237 0.6185 0.5110 0.6451 0.5691 0.2969 0.15840.2783 −0.1109 0.2284 0.5913 0.4924 0.6369 0.5879 0.2495 0.1444 0.38910.0000 0.2304 0.4980 0.5014 0.5993 0.5906 0.2076 0.1300 0.5000 0.11090.2294 0.3702 0.4924 0.5511 0.5825 0.1754 0.1167 0.6109 0.2217 0.22490.2468 0.5110 0.5055 0.5609 0.1539 0.1055 0.7217 0.3326 0.2160 0.15490.4992 0.4648 0.5182 0.1418 0.0973 0.8326 0.4434 0.2048 0.1010 0.45500.4232 0.4513 0.1367 0.0924 0.9434 0.5543 0.2000 0.0758 0.3867 0.37410.3672 0.1358 0.0908 1.0543 0.6651 0.2173 0.0650 0.3082 0.3154 0.28150.1363 0.0917 1.1651 0.7760 0.2727 0.0588 0.2327 0.2511 0.2095 0.13620.0940 1.2760 0.8868 0.3701 0.0535 0.1694 0.1882 0.1577 0.1347 0.09621.3868 0.9977 0.4907 0.0491 0.1219 0.1338 0.1228 0.1325 0.0992 1.49771.1085 0.5962 0.0458 0.0896 0.0918 0.0971 0.1305 0.1087 1.6085

B) Effect of Pupil Size

FIGS. 41 to 43 show the variation in through focus RIQ with pupil sizefor Iteration B1, Iteration B2 and Iteration B3 respectively. Each powerprofile is relatively stable, in that the RIQ retains the combination ofa relatively high RIQ (in comparison to, for example, a centre distanceaspheric multifocal) in combination with a relatively long through focusrange (in comparison to, for example, a centre near asphericmultifocal). FIGS. 44 and 45 show the variation in through focus RIQwith pupil size for the two concentric bifocals and two asphericmultifocals, respectively. From these figures it can be seen that,comparatively, the change in RIQ and through focus RIQ performance isless stable for these lenses than Iteration B1, Iteration B2 andIteration B3.

C) Monocular Design

As described above, Iteration B2 can provide an RIQ of 0.40 or abovefrom distance vision to about an intermediate vergence of about 1.1Diopters. When appropriate level of defocus is added to the sameiteration while correcting the other eye, through-focus RIQ can beextended from 1.1 Diopters to up close, say 2.20D target vergence, i.e.binocularly combined the candidate eye may maintain an RIQ of 0.40 orabove from distance test distance to all the way up close to 2.2Diopters. Using this monocular design approach and assuming therecipient accepts the monocular design, the combined through focusperformance is substantially extended.

Referring to the through focus profiles shown in FIGS. 46 and 47 , whichare described herein below, under the monocular design approach, onelens will be selected to have a base power that shifts the through focuscurve to the extreme left (starting at −2.50D mark) and the other lensselected to have a base power that shifts the through focus curveslightly to the left (starting at −1.50D mark).

FIGS. 46 and 47 show the through-focus RIQ of the design of two pairs ofpower profiles (Binocular ‘Q’ correction). Each lens in the pair hasbeen designed to extend RIQ in combination with the other lens in thepair. The defocus and higher order spherical aberration coefficients forthese combinations are specified in Tables 7 and 8 respectively.

TABLE 7 Defocus and higher order spherical aberration coefficients offirst exemplary embodiment for monocular design of lenses for presbyopia(Effective add of 1.50D in the negative direction of through-focuscurve) Combination C (2,0) C(4,0) C(6,0) C(8,0) C(10,0) C(12,0) C(14,0)C(16,0) C(18,0) C(20,0) Right Eye 0.28 −0.100 0.025 0.075 0.025 0.0250.025 0.025 0.025 0.000 Left Eye 0.57 0.125 −0.075 −0.075 −0.025 0.0000.025 0.025 −0.025 −0.025

TABLE 8 Defocus and higher order spherical aberration coefficients ofsecond exemplary embodiment for monocular design of lenses forpresbyopia (Effective add of 2.50D in the negative direction ofthrough-focus curve) Combination C (2,0) C(4,0) C(6,0) C(8,0) C(10,0)C(12,0) C(14,0) C(16,0) C(18,0) C(20,0) Right Eye 0.433 −0.100 −0.0500.025 0.025 −0.025 −0.025 0.000 0.000 0.000 Left Eye 0.866 −0.100 −0.0500.025 0.025 −0.025 −0.025 0.000 0.000 0.000The power profiles described in relation to Table 7 and Table 8 areexamples of combinations of higher order aberrations that provideenhanced through-focus performance on the negative side of thethrough-focus function. Similarly, using this monocular design approach,the combined through-focus performance can also be substantiallyextended on the right side of the through-focus function, provided anappropriate level of defocus is added to a selected combination ofhigher order aberrations. FIGS. 48 and 49 show examples with arelatively constant RIQ (>0.35) over a range of defocus, in the positivedirection of the through-focus function. The defocus and higher orderspherical aberration coefficients for these combinations are specifiedin Tables 9 and 10, respectively.

TABLE 9 Defocus and higher order spherical aberration coefficients ofthird exemplary embodiment for monocular design of lenses for presbyopia(Effective add of 1.50D in the positive direction of through-focuscurve) Combination C (2,0) C(4,0) C(6,0) C(8,0) C(10,0) C(12,0) C(14,0)C(16,0) C(18,0) C(20,0) Right Eye −0.28 −0.125 −0.050 0.075 0.025 −0.0250.000 0.000 0.000 0.000 Left Eye −0.43 −0.125 −0.050 0.075 0.025 −0.0250.000 0.000 0.000 0.000

TABLE 10 Defocus and higher order spherical aberration coefficients offourth exemplary embodiment for monocular design of lenses forpresbyopia (Effective add of 2.50D in the positive direction ofthrough-focus curve) Combination C (2,0) C(4,0) C(6,0) C(8,0) C(10,0)C(12,0) C(14,0) C(16,0) C(18,0) C(20,0) Right Eye −0.43 −0.125 −0.0500.075 0.025 −0.025 0.000 0.000 0.000 0.000 Left Eye −0.86 −0.125 −0.0500.075 0.025 −0.025 0.000 0.000 0.000 0.000

10. Design for Peripheral Field

In some embodiments, when selecting a combination of HOA to form a powerprofile, the weight given to peripheral vision may be increased. Thismay, for example, be applicable when the recipient plays certain sportsin which peripheral vision is important.

FIG. 50 shows a graph of RIQ (again Visual Strehl Ratio), for threedifferent power profiles that substantially equalise RIQ across thehorizontal visual field. The RIQ measures were obtained for a 5 mmpupil. The defocus and higher order spherical aberration coefficientsfor each power profile are shown in Table 11.

TABLE 11 Defocus and higher order spherical aberration coefficients ofthree exemplary embodiments for substantially constant RIQ over extendedhorizontal field angles Iteration C(2,0) C(4,0) C(6,0) C(8,0) C(10,0)C(12,0) C(14,0) C(16,0) C(18,0) C(20,0) Iteration −1.506 0.111 −0.040−0.015 0.007 0.025 0.011 −0.025 −0.003 0.017 A3 Iteration −1.504 0.114−0.037 −0.013 0.009 0.027 0.013 −0.024 −0.002 0.016 A4 Iteration −1.5010.117 −0.034 −0.010 0.012 0.029 0.014 −0.023 −0.002 0.015 A5

Each of Iteration A3, Iteration A4 and Iteration A5 produced an on-axisRIQ of about 0.50 across zero to 30 degrees field angle (if horizontalsymmetry is assumed, that is 60 degrees in total across both nasal andtemporal fields). The RIQ on-axis is also about 0.50, which is lowerthan some other embodiments where degradation in RIQ below 0.50 withincreasing field angle is permitted.

Accordingly, in further embodiments, the RIQ on-axis may be traded-offagainst RIQ at high field angles. For example, RIQ may be permitted todrop to 0.20 at 30 degrees field angle (but remain at 0.50 or above for20 degrees field angle and less), to allow a selection of HOA thatincreases on-axis RIQ above those shown in FIG. 50 . Power profiledesigns for peripheral vision may be selected for a lens designed toprovide a slope of RIQ (providing stimulus to retard or encourage eyegrowth under the optical feedback mechanism explanation foremmetropisation), or correction/lenses for presbyopia (emmetropic,myopic or hyperopic) or for other eyes.

11. Selection of Positive and Negative Phase

For any particular recipient of a lens, device or a method disclosedherein, a selection may be made between any two power profiles ofopposite phases. In this context, the term ‘opposite phase’ identifiespower profiles that have identical magnitudes of specific combinationsets of higher order aberrations over a desired pupil, while their signsare opposite to each other. FIGS. 51 and 52 show power profileiterations E1 and E2, which are examples of power profiles with oppositephases. Table 12 reflects the magnitudes and signs of the higher orderspherical aberration terms for iterations E1 and E2.

The lenses of opposite phase described herein may result in the sameon-axis peak RIQ. The through focus RIQ performance of such phaseprofile pairs may be mirror images of each other across the Y-axis (i.e.shifted apart by defocus), as shown in FIG. 53 . However, this wouldresult only if the inherent higher order aberration profile isnegligibly small (say for example primary spherical aberration in therange of −0.02 μm to 0.02 μm over a 5 mm pupil).

TABLE 12 Defocus and higher order spherical aberration coefficients oftwo exemplary embodiments with opposite phases (i.e. mirror imaged powerprofiles across the X-axis). Iteration C(2,0) C(4,0) C(6,0) C(8,0)C(10,0) C(12,0) C(14,0) C(16,0) C(18,0) C(20,0) Iteration −2.015 −0.1020.021 0.019 0.025 0.010 −0.025 −0.006 0.016 −0.003 E1 Iteration −1.5730.102 −0.021 −0.019 −0.025 −0.010 0.025 0.006 −0.016 0.003 E2

The interactions between the inherent aberration profiles of thecandidate eyes and a selected phase profile may either have a) animproved or b) degraded effect on the objective/subjectiveoptical/visual performance. As the through-focus RIQ is dependent on theinherent aberration profile, a phase profiles selected for instance maybe useful to change the slope of through-focus RIQ in the direction thatwould favour the emmetropisation process for myopic or hyperopic eyes;or alternatively the same phase profile could be used to mitigate thepresbyopic symptoms in alternative candidate eyes.

FIGS. 54 and 55 show how the through-focus RIQ of opposite phaseprofiles are dependent on the inherent ocular aberration of thecandidate eye (in this example positive spherical aberration).Accordingly, embodiments of the invention involve providing lenses ofthe same design, but opposite phase and allowing the recipient to selectthe preferred phase. The process of selection can be via an objectiveassessment of through-focus RIQ performance metric or could be purely asubjective preference via visually guided tests.

12. Combination Identification and Selection

As described above, it is possible to provide a desirable on-axis RIQfor distance and appropriate through focus RIQ that would enable bettervisual performance for intermediate and near vergences by choosing anappropriate combination of HOA. This combination of higher orderaberrations may contain a correction for the inherent aberration profileof the test candidate. The Appendix A to this specification lists 78combinations of higher order spherical aberration coefficients thatprovide both a usefully high RIQ and an option to provide an extendedthrough focus RIQ in the negative direction (left hand side). Also shownin the Appendix A, as a point of comparison, is a combination which doesnot have any spherical aberration, of any order. The Appendix B showsthe through-focus RIQ values for the combinations listed in the AppendixA. All calculations were performed for a pupil size of 4 mm, however theapproach can be extended to any other appropriate/desired pupil sizes ifrequired.

The through-focus RIQ measures of the 78 aberration combinations areshown in FIG. 56 , the black line showing the symmetrical RIQ that hasresulted from a combination that has no higher order aberrations andwhile the lighter lines (i.e. gray lines) showing the enhancedperformance in the negative direction of the through-focus RIQ function,for the 78 combinations that involve higher order spherical aberrationterms.

From FIG. 56 , a number of observations can be made. All of the 78profiles with higher order spherical aberration terms provide anextended through focus performance in the negative direction,particularly when an appropriate selection of a negative power is madeto shift the plotted through-focus profile towards negative defocus(left). All of the 78 profiles include a range over which RIQ is 0.10 orhigher of at least 2 Diopters. Several of the 78 profiles include arange over which RIQ is 0.10 or higher of at least 2.25 Diopters. All ofthe 78 profiles include an RIQ (Visual Strehl Ratio—monochromatic) thatpeaks above 0.35. Many of the profiles include an RIQ that peaks abovethe thresholds of 0.40, 0.50, 0.60 and 0.70 and some combinations resultin a peak that lies above 0.80 mark.

The spherical aberration terms vary in the combinations, from one(example: combination 77) through to all nine. In other embodiments evenhigher orders of spherical aberration terms may be added, to createadditional combinations.

The combination 77 in the appendix A introduces only primary sphericalaberration. Primary spherical aberration was proposed in the U.S. Pat.No. 6,045,578 (Collins and Wildsoet) for progressing myopia. Thecombination 77 shows that by selecting a particular level of primaryspherical aberration, the aberration profile may be beneficially usedfor a presbyopic eye. In addition, considering the application ofmyopia, when combination 77 (or any other embodiment including only PSA)is set with a focal length matching the existing myopia, then theon-axis through focus RIQ is substantially neutral, in that the peak RIQis placed on the retina. Placing the peak RIQ on the retina would accordwith a traditional design approach as taught by Collins. In contrast, astimulus to retard eye growth on-axis under the optical feedbackexplanation of emmetropisation is achieved if the retina is located onthe negative side of the graph shown in FIG. 57 (i.e. the focal lengthof the lens is longer than the eye). In other words, the aberrationprofile will include a C(2,0) term with further negative power over theamount required to correct myopia.

Appendix C to this specification lists another 67 combinations of higherorder coefficients that provide both a usefully high RIQ and an optionto provide an extended through-focus RIQ in the positive direction(right hand side). Also shown in Appendix C, as a point of comparison,is a combination which does not have any spherical aberration of anyorder. The Appendix D shows the through-focus RIQ values for thecombinations listed in Appendix C. Again, all calculations wereperformed for a pupil size of 4 mm, however the approach can be extendedto any other appropriate/desired pupil sizes, if required.

The through-focus RIQ measures of the 67 aberration combinations areshown in the FIG. 58 , the black line showing the symmetrical RIQ thathas resulted from a combination that has no higher order aberrations andwhile the lighter (i.e. gray lines) showing the enhanced performance inthe positive direction of the through-focus RIQ function, for the 67combinations that involved higher order spherical aberration terms.

From the FIG. 58 , a number of observations can be made. All of the 67profiles with higher order spherical aberration terms provide anextended through-focus performance in the positive directionparticularly when appropriate selection of a negative power is made toshift the plotted through-focus profile towards negative defocus (left).All of the 67 profiles include a range over which the RIQ is 0.10 orhigher of greater than 2.50D.

FIG. 59 shows an example workflow diagram for identifying a powerprofile for application to a presbyopic eye.

13. Spherical Aberration and Astigmatism

In the previous sections iterations B1, B2 and B3 were described foremmetropic presbyopia. When considering the astigmatic presbyopia, twodifferent methods can be adopted. A first method of correction iscompleted by considering astigmatic refractive error as an equivalentsphere. In this method, the spherical equivalent prescription is deducedby dividing the cylindrical/astigmatic power divided two (S=−C/2). Thisis a very common approach often considered to address low to moderateamounts of astigmatism, say up to −1.50D. Once the equivalent sphere isavailed, the same iterations described herein, say for example B1, B2 orB3 can be used as an effective prescription, once the defocus term isadjusted to suit the spherical equivalent.

A second method considers preparation of a toric prescription for bothastigmatism and presbyopia. FIG. 60 shows an exemplary embodiment thatincludes a toric power profile to treat both astigmatism and presbyopia.In this case, the prescription is made to correct an individual who hasan astigmatic correction of −1.00D @ 90 and requires an additional powerto enable near viewing. As can be noted from the figure, the differencebetween the horizontal and vertical meridian is −1.00D, this magnitudeis set to corrects the astigmatism in the above test case; while thehigher order spherical aberration combination is aimed to mitigate thepresbyopic symptoms.

14. Implementation

Aberration profiles of the types described herein above may beimplemented in a number of lenses, ocular devices and as methods.

For example, contact lenses (hard or soft), corneal onlays, cornealinlays, and lenses for intraocular devices (both anterior and posteriorchamber) may all include the combination aberration profiles discussed.Techniques to design lenses and to achieve a power profile are known andwill are not described herein in any detail.

The aberration profiles can be applied to spectacle lenses. However,because the aberration profiles require alignment of the eye with thecentre of the optics providing the aberration profile, then benefit mayonly be apparent for one particular direction of gaze. Recentlyelectro-active lenses have been proposed that can track the direction ofgaze and change the refractive properties of the lenses in response.Using electro-active lenses the aberration profile can move with theeye, which may increase the utility of the disclosed aberration profilesfor spectacle lenses.

The aberration profile may be provided on a lens of an intraocular lens.In some embodiments the intraocular lens may include haptics thatprovide for accommodation. In other embodiments the lens may have afixed focal length. The aberration profile may be provided on asupplementary endo-capsular lens.

The disclosed aberration profiles may be provided to an eye throughcomputer-assisted surgery. For example refractive surgery or cornealablation may be used to form a selected aberration profile. The powerprofile or required change in corneal shape is determined and input tothe laser system (LASIK or LASEK) for application to the eye of thepatient.

Where the aberration profiles are to be included in a lens, then theaberration profile may first be translated into a lens thickness profilefor input to computer assisted manufacturing. Taking for example, thelens power profile D1 shown in FIG. 61 , which is a combination ofZernike higher order spherical aberration terms, is converted to anaxial thickness profile for a contact lens, taking account of therefractive index of the contact lens material (in this case, contactlens material refractive index of 1.420). An example thickness profileis shown in FIG. 62 . Features of the power/thickness profiles caneither be put on the front or the back surface or a combination of both,under consideration of the refractive indices of lens and cornea. Onceall the parameters i.e. the thickness profile, power profile, backsurface shape, diameter and refractive index of the material have beendetermined, then this is input to a computer assisted lathe to producethe contact lens. Similar approaches can be adopted for other lenses.

The aberration profile may be selected and identified as a custom lensfor an individual. The process for design of the aberration profileincludes measuring the wavefront aberration of the eye and designing anaberration profile to achieve a through focus RIQ profile describedherein. The design process includes identifying the spherical aberrationin the natural eye only and designing an aberration profile for thelens, device or method that, in combination with the sphericalaberration of the eye provides a required RIQ profile. As describedherein above, the required RIQ profile may differ depending on theapplication of the lens—as different requirements may apply between aperson with progressing myopia and a person with presbyopia. In someembodiments other aberrations in the eye, for example astigmatism, comaor trefoil are ignored. In other embodiments, these are taken intoaccount. For example, as described above, the presence of astigmatismaffects the combinations of aberrations that provide a through focus RIQthat inhibits eye growth under the optical feedback explanation ofemmetropisation. In other embodiments these aberrations are incorporatedinto the design. For example, when producing a lens design, a base lensmay be produced that corrects for any defocus and corrects one or moreof astigmatism, coma and trefoil. On top of this base profile isprovided a spherical aberration profile designed to achieve (in thesense of using as an objective design) the profiles described herein.The spherical aberration profile may be selected using a trial and errorapproach, for example by identifying a candidate profile, computing thethrough focus RIQ and evaluating whether the through focus RIQ has anacceptable profile.

In another approach aberration profiles may be designed for populationaverages. One approach for designing population average lenses is tonormalise the design for pupil size.

The description of the aberration profiles for lenses, devices andmethods has been provided by way of mathematical explanation. Thisallows for precision in describing the aberration profiles. However,lenses, devices and methods will not have such precision. For exampletolerances and inaccuracies arising during manufacture will result invariations of the lens profile. The approximate power profile of a lenscan be measured using a wavefront aberrometer. From this an approximatemeasure of through focus RIQ, for example Visual Strehl Ratio, can bedetermined.

It will be understood that the invention disclosed and defined in thisspecification extends to all alternative combinations of two or more ofthe individual features mentioned or evident from the text or drawings.All of these different combinations constitute various alternativeaspects of the invention.

15. Appendix A—Example Combinations of Spherical Aberration

Combination C (2,0) C(4,0) C(6,0) C(8,0) C(10,0) C(12,0) C(14,0) C(16,0)C(18,0) C(20,0) No Aberr 0 0 0 0 0 0 0 0 0 0 1 0 −0.125 −0.075 0.0000.000 0.000 0.000 0.000 0.000 0.000 2 0 −0.100 −0.075 0.000 0.000 0.0000.000 0.000 0.000 0.000 3 0 −0.100 −0.025 0.025 0.000 0.000 0.000 0.0000.000 0.000 4 0 −0.100 0.025 0.075 0.025 0.025 0.025 0.025 0.025 0.000 50 −0.075 −0.075 0.000 0.000 0.000 0.000 0.000 0.000 0.000 6 0 −0.075−0.025 0.050 0.000 −0.025 −0.025 0.000 0.025 0.000 7 0 −0.050 −0.0750.000 0.000 0.000 0.000 0.000 0.000 0.000 8 0 −0.050 −0.050 0.050 0.0250.000 0.000 0.000 0.000 0.000 9 0 −0.050 −0.025 0.050 0.000 −0.025−0.025 0.000 0.025 0.025 10 0 −0.025 −0.075 0.000 0.000 0.000 0.0000.000 0.000 0.000 11 0 −0.025 −0.025 0.050 0.025 −0.025 −0.025 0.0000.025 0.025 12 0 0.000 −0.075 0.000 0.000 0.000 0.000 0.000 0.000 0.00013 0 0.000 −0.075 0.050 0.025 0.000 0.025 0.000 −0.025 0.000 14 0 0.000−0.050 0.000 −0.025 −0.025 0.025 0.025 −0.025 −0.025 15 0 0.000 −0.0500.050 0.025 −0.025 −0.025 −0.025 0.000 0.025 16 0 0.000 −0.025 0.0750.000 −0.025 0.025 0.025 0.025 0.025 17 0 0.025 −0.075 0.000 −0.025−0.025 0.025 0.025 0.000 0.000 18 0 0.025 −0.075 0.000 0.000 0.000 0.0000.000 0.000 0.000 19 0 0.025 −0.075 0.025 0.025 −0.025 −0.025 −0.0250.000 0.025 20 0 0.025 −0.075 0.050 0.025 −0.025 −0.025 −0.025 0.0000.000 21 0 0.025 −0.050 0.000 0.000 0.000 0.000 0.000 0.000 0.000 22 00.025 −0.050 0.050 0.000 −0.025 −0.025 0.000 0.025 0.025 23 0 0.025−0.050 0.050 0.025 0.000 0.000 −0.025 −0.025 0.000 24 0 0.025 −0.0250.075 0.000 −0.025 0.025 0.025 0.025 0.025 25 0 0.050 −0.075 0.000 0.000−0.025 0.000 0.000 0.025 0.025 26 0 0.050 −0.075 0.000 0.000 0.000 0.0000.000 0.000 0.000 27 0 0.050 −0.075 0.025 0.025 −0.025 0.000 0.000−0.025 0.000 28 0 0.050 −0.075 0.025 0.025 −0.025 0.000 0.000 0.0250.025 29 0 0.050 −0.075 0.025 0.025 0.000 0.000 −0.025 −0.025 0.000 30 00.050 −0.075 0.025 0.025 0.000 0.025 0.025 0.025 0.025 31 0 0.050 −0.0500.000 0.000 0.000 0.000 0.000 0.000 0.000 32 0 0.050 −0.025 −0.025−0.025 −0.025 0.025 0.025 0.000 −0.025 33 0 0.050 −0.025 0.075 0.025−0.025 0.025 0.025 0.025 0.025 34 0 0.075 0.050 −0.025 −0.025 0.0000.000 0.000 0.000 0.000 35 0 0.075 −0.075 −0.025 −0.025 0.000 0.0250.000 0.000 0.000 36 0 0.075 −0.075 −0.025 0.000 0.000 0.025 0.025 0.0000.000 37 0 0.075 −0.075 0.000 0.000 −0.025 −0.025 0.000 0.000 0.000 38 00.075 −0.075 0.000 0.000 −0.025 0.000 0.000 0.000 0.000 39 0 0.075−0.075 0.000 0.000 0.000 0.000 0.000 0.000 0.000 40 0 0.075 −0.075 0.0000.025 −0.025 −0.025 0.000 0.000 0.000 41 0 0.075 −0.075 0.000 0.025−0.025 0.000 0.000 0.000 0.000 42 0 0.075 −0.050 −0.050 −0.025 0.0000.000 0.025 0.000 −0.025 43 0 0.075 −0.050 0.000 0.000 0.000 0.000 0.0000.000 0.000 44 0 0.075 −0.025 0.000 0.000 0.000 0.000 0.000 0.000 0.00045 0 0.075 −0.025 0.050 0.000 −0.025 0.025 0.025 0.000 0.000 46 0 0.100−0.075 −0.050 −0.025 0.000 0.025 0.025 −0.025 −0.025 47 0 0.100 −0.075−0.050 0.000 0.000 0.025 0.025 −0.025 −0.025 48 0 0.100 −0.075 −0.0250.000 0.000 0.000 0.000 0.000 0.000 49 0 0.100 −0.075 −0.025 0.000 0.0000.025 0.000 0.000 0.000 50 0 0.100 −0.075 0.000 0.000 0.000 0.000 0.0000.000 0.000 51 0 0.100 −0.075 0.000 0.025 −0.025 −0.025 0.025 0.0250.000 52 0 0.100 −0.050 −0.050 −0.025 0.000 −0.025 −0.025 −0.025 −0.02553 0 0.100 −0.050 −0.025 −0.025 −0.025 0.025 0.000 −0.025 0.000 54 00.100 −0.050 0.000 0.000 0.000 0.000 0.000 0.000 0.000 55 0 0.100 −0.0500.000 0.000 0.000 0.025 0.025 0.000 0.000 56 0 0.100 −0.050 0.000 0.0000.000 0.025 0.025 0.025 0.025 57 0 0.100 −0.050 0.000 0.025 0.025 0.000−0.025 −0.025 −0.025 58 0 0.100 −0.025 0.000 0.000 0.000 0.000 0.0000.000 0.000 59 0 0.100 −0.025 0.000 0.025 0.025 0.000 −0.025 −0.025−0.025 60 0 0.100 −0.025 0.025 −0.025 −0.025 0.025 0.025 0.000 0.000 610 0.100 0.000 0.000 −0.025 0.000 0.025 0.000 0.000 0.025 62 0 0.1000.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 63 0 0.100 0.000 0.0500.000 −0.025 0.025 0.000 −0.025 0.000 64 0 0.125 −0.075 −0.075 −0.0250.000 0.025 0.025 −0.025 −0.025 65 0 0.125 −0.075 −0.075 0.000 0.0000.000 0.000 0.000 0.000 66 0 0.125 −0.075 0.000 0.000 0.000 0.000 0.0000.000 0.000 67 0 0.125 −0.050 −0.025 −0.025 −0.025 0.000 0.000 0.0000.000 68 0 0.125 −0.050 −0.025 −0.025 −0.025 0.025 0.000 0.000 0.000 690 0.125 −0.050 −0.025 0.000 0.000 0.025 0.025 0.000 0.000 70 0 0.125−0.050 0.000 0.000 0.000 0.000 0.000 0.000 0.000 71 0 0.125 −0.050 0.0000.025 0.025 0.025 0.000 0.000 0.000 72 0 0.125 −0.025 0.000 −0.025−0.025 0.000 0.000 −0.025 −0.025 73 0 0.125 −0.025 0.000 0.000 0.0000.000 0.000 0.000 0.000 74 0 0.125 −0.025 0.025 0.000 −0.025 0.000 0.0000.000 0.000 75 0 0.125 −0.025 0.025 0.000 0.000 0.025 0.025 0.000 0.00076 0 0.125 −0.025 0.025 0.025 0.025 −0.025 0.025 0.025 0.025 77 0 0.1250.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 78 0 0.125 0.000 0.025−0.025 −0.025 0.025 0.000 −0.025 −0.025

16. Appendix B—Through Focus RIQ for Combinations of SphericalAberration in Appendix A

Combination −1.50 −1.25 −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.001.25 1.50 No Aberr 0.024 0.040 0.073 0.148 0.307 0.709 1.000 0.709 0.3070.148 0.073 0.040 0.024 1 0.089 0.135 0.192 0.243 0.304 0.434 0.6060.667 0.542 0.329 0.152 0.056 0.021 2 0.084 0.131 0.196 0.265 0.3460.482 0.643 0.676 0.514 0.281 0.113 0.036 0.012 3 0.028 0.053 0.1150.258 0.473 0.628 0.648 0.595 0.479 0.310 0.161 0.071 0.028 4 0.0390.067 0.153 0.313 0.458 0.493 0.477 0.492 0.470 0.361 0.220 0.112 0.0525 0.082 0.128 0.198 0.281 0.384 0.532 0.675 0.675 0.481 0.236 0.0800.021 0.006 6 0.100 0.129 0.157 0.246 0.402 0.514 0.542 0.559 0.5150.338 0.146 0.051 0.024 7 0.083 0.129 0.199 0.289 0.412 0.576 0.7040.666 0.445 0.196 0.054 0.010 0.002 8 0.069 0.105 0.176 0.305 0.4790.603 0.614 0.565 0.454 0.262 0.099 0.030 0.010 9 0.124 0.168 0.1810.212 0.338 0.502 0.579 0.579 0.508 0.319 0.117 0.027 0.016 10 0.0890.133 0.201 0.293 0.425 0.607 0.730 0.656 0.409 0.161 0.034 0.003 0.00111 0.104 0.159 0.199 0.247 0.359 0.508 0.581 0.570 0.502 0.326 0.1250.035 0.023 12 0.098 0.141 0.206 0.293 0.423 0.618 0.749 0.649 0.3770.134 0.021 0.001 0.002 13 0.157 0.206 0.250 0.282 0.354 0.482 0.5420.480 0.364 0.232 0.120 0.060 0.032 14 0.092 0.184 0.314 0.371 0.3900.505 0.592 0.481 0.297 0.204 0.161 0.097 0.041 15 0.153 0.215 0.2470.261 0.324 0.453 0.533 0.514 0.447 0.307 0.129 0.038 0.025 16 0.1520.207 0.237 0.260 0.363 0.509 0.531 0.442 0.363 0.265 0.137 0.056 0.02917 0.158 0.218 0.286 0.308 0.324 0.457 0.611 0.564 0.352 0.181 0.1010.048 0.011 18 0.111 0.152 0.213 0.293 0.410 0.604 0.754 0.650 0.3560.113 0.013 0.004 0.004 19 0.168 0.205 0.235 0.285 0.367 0.476 0.5390.482 0.365 0.253 0.138 0.052 0.023 20 0.161 0.202 0.237 0.282 0.3610.468 0.518 0.465 0.378 0.267 0.124 0.038 0.019 21 0.081 0.116 0.1740.255 0.405 0.680 0.878 0.715 0.342 0.093 0.015 0.002 0.001 22 0.1510.212 0.253 0.256 0.304 0.463 0.584 0.514 0.360 0.223 0.095 0.016 0.00323 0.153 0.205 0.242 0.255 0.316 0.493 0.638 0.563 0.363 0.201 0.0960.041 0.023 24 0.159 0.214 0.250 0.256 0.322 0.476 0.548 0.465 0.3570.251 0.127 0.046 0.021 25 0.158 0.201 0.231 0.253 0.312 0.472 0.6480.612 0.359 0.141 0.075 0.067 0.043 26 0.126 0.166 0.222 0.293 0.3880.567 0.739 0.657 0.350 0.099 0.008 0.005 0.006 27 0.161 0.203 0.2360.253 0.304 0.475 0.648 0.593 0.370 0.190 0.091 0.039 0.015 28 0.1640.201 0.226 0.253 0.323 0.472 0.604 0.547 0.352 0.197 0.112 0.058 0.03129 0.171 0.206 0.240 0.274 0.328 0.463 0.608 0.564 0.362 0.193 0.0940.036 0.012 30 0.171 0.206 0.231 0.259 0.326 0.475 0.626 0.589 0.3630.150 0.057 0.031 0.015 31 0.097 0.135 0.192 0.268 0.389 0.628 0.8480.728 0.347 0.078 0.006 0.001 0.003 32 0.074 0.134 0.238 0.370 0.4620.553 0.624 0.516 0.286 0.156 0.129 0.096 0.052 33 0.159 0.212 0.2450.251 0.305 0.461 0.564 0.496 0.375 0.264 0.138 0.048 0.019 34 0.0220.044 0.114 0.279 0.496 0.623 0.634 0.591 0.479 0.310 0.160 0.069 0.03035 0.161 0.200 0.244 0.318 0.404 0.493 0.584 0.550 0.352 0.162 0.0720.032 0.009 36 0.151 0.217 0.289 0.353 0.390 0.455 0.568 0.563 0.3730.173 0.080 0.042 0.013 37 0.151 0.206 0.264 0.304 0.336 0.450 0.6300.628 0.372 0.127 0.038 0.014 0.004 38 0.164 0.211 0.254 0.279 0.3090.455 0.681 0.686 0.400 0.126 0.027 0.011 0.005 39 0.142 0.181 0.2320.292 0.364 0.512 0.699 0.664 0.364 0.097 0.005 0.006 0.008 40 0.1550.222 0.286 0.331 0.369 0.465 0.601 0.579 0.365 0.172 0.085 0.037 0.00841 0.151 0.204 0.251 0.282 0.320 0.459 0.661 0.659 0.405 0.163 0.0620.031 0.018 42 0.118 0.171 0.252 0.367 0.460 0.506 0.539 0.496 0.3290.166 0.098 0.069 0.035 43 0.115 0.156 0.212 0.283 0.376 0.563 0.7840.729 0.371 0.080 0.001 0.003 0.005 44 0.086 0.126 0.186 0.272 0.3920.602 0.826 0.761 0.391 0.094 0.012 0.005 0.001 45 0.153 0.203 0.2570.284 0.316 0.452 0.609 0.566 0.367 0.207 0.104 0.035 0.011 46 0.1800.256 0.316 0.408 0.497 0.493 0.427 0.336 0.212 0.122 0.109 0.104 0.06447 0.171 0.253 0.325 0.407 0.458 0.443 0.429 0.400 0.289 0.173 0.1310.112 0.066 48 0.151 0.211 0.281 0.358 0.417 0.470 0.566 0.585 0.3970.155 0.035 0.004 0.004 49 0.155 0.203 0.255 0.330 0.407 0.472 0.5600.561 0.375 0.168 0.075 0.042 0.018 50 0.159 0.197 0.240 0.289 0.3390.449 0.636 0.663 0.396 0.110 0.005 0.007 0.009 51 0.185 0.272 0.3600.392 0.353 0.357 0.461 0.486 0.330 0.168 0.108 0.077 0.037 52 0.0960.141 0.222 0.351 0.472 0.508 0.515 0.524 0.412 0.196 0.057 0.024 0.02153 0.158 0.206 0.242 0.306 0.392 0.462 0.534 0.533 0.381 0.208 0.1160.063 0.025 54 0.134 0.177 0.231 0.296 0.365 0.494 0.694 0.710 0.4090.101 0.001 0.004 0.007 55 0.152 0.204 0.259 0.316 0.366 0.464 0.6260.630 0.369 0.110 0.031 0.028 0.016 56 0.161 0.207 0.253 0.290 0.3380.458 0.619 0.607 0.360 0.117 0.033 0.027 0.022 57 0.143 0.197 0.2680.357 0.426 0.471 0.522 0.486 0.298 0.128 0.086 0.078 0.044 58 0.1050.151 0.214 0.299 0.398 0.542 0.721 0.717 0.423 0.123 0.017 0.003 0.00359 0.110 0.169 0.259 0.371 0.457 0.518 0.571 0.515 0.302 0.113 0.0680.073 0.053 60 0.158 0.202 0.246 0.308 0.374 0.455 0.553 0.536 0.3660.196 0.093 0.030 0.008 61 0.118 0.160 0.205 0.284 0.407 0.520 0.5880.569 0.421 0.224 0.088 0.026 0.007 62 0.076 0.119 0.189 0.297 0.4370.593 0.722 0.683 0.425 0.165 0.053 0.021 0.006 63 0.156 0.207 0.2430.258 0.318 0.460 0.563 0.511 0.364 0.236 0.140 0.075 0.044 64 0.1940.280 0.335 0.402 0.502 0.516 0.402 0.272 0.179 0.124 0.113 0.113 0.08665 0.155 0.251 0.353 0.432 0.463 0.418 0.355 0.368 0.387 0.303 0.1630.062 0.021 66 0.175 0.210 0.246 0.284 0.316 0.385 0.554 0.643 0.4390.141 0.009 0.008 0.010 67 0.163 0.214 0.265 0.328 0.402 0.466 0.5290.536 0.389 0.186 0.072 0.031 0.009 68 0.163 0.201 0.232 0.294 0.3970.476 0.522 0.506 0.365 0.192 0.103 0.062 0.031 69 0.157 0.220 0.2810.355 0.428 0.468 0.519 0.533 0.375 0.160 0.065 0.050 0.032 70 0.1530.198 0.248 0.304 0.354 0.431 0.590 0.664 0.449 0.143 0.010 0.005 0.00871 0.153 0.201 0.261 0.343 0.412 0.458 0.535 0.552 0.372 0.143 0.0510.040 0.024 72 0.151 0.207 0.259 0.316 0.391 0.466 0.517 0.487 0.3530.210 0.114 0.042 0.006 73 0.126 0.176 0.241 0.320 0.401 0.489 0.6090.645 0.446 0.168 0.033 0.005 0.004 74 0.161 0.203 0.237 0.270 0.3330.456 0.608 0.618 0.406 0.179 0.081 0.038 0.010 75 0.159 0.202 0.2430.289 0.349 0.456 0.592 0.584 0.367 0.145 0.046 0.010 0.003 76 0.0760.148 0.260 0.351 0.375 0.411 0.515 0.518 0.321 0.134 0.082 0.053 0.00877 0.096 0.147 0.224 0.329 0.451 0.554 0.619 0.595 0.422 0.202 0.0740.027 0.007 78 0.160 0.216 0.272 0.318 0.372 0.434 0.455 0.411 0.3440.276 0.169 0.060 0.018

17. Appendix C—Example Combinations of Spherical Aberration

Combination C (2,0) C(4,0) C(6,0) C(8,0) C(10,0) C(12,0) C(14,0) C(16,0)C(18,0) C(20,0) No Aberr 0 0 0 0 0 0 0 0 0 0 101 0 −0.125 −0.075 0.0000.025 −0.025 −0.025 0.025 0.000 −0.025 102 0 −0.125 −0.050 0.000 0.0250.000 −0.025 0.025 0.000 −0.025 103 0 −0.125 −0.050 0.000 0.025 0.000−0.025 0.025 0.025 −0.025 104 0 −0.125 −0.050 0.025 0.025 −0.025 −0.0250.025 0.000 −0.025 105 0 −0.125 −0.050 0.050 0.025 −0.025 0.000 0.025−0.025 −0.025 106 0 −0.125 −0.050 0.050 0.025 −0.025 0.025 0.000 0.0000.025 107 0 −0.125 −0.025 −0.025 0.025 0.025 −0.025 0.000 0.025 0.000108 0 −0.125 −0.025 0.000 0.000 0.025 −0.025 −0.025 0.025 0.025 109 0−0.125 −0.025 0.000 0.000 0.025 0.000 −0.025 0.025 0.025 110 0 −0.125−0.025 0.000 0.025 0.025 −0.025 −0.025 0.025 0.000 111 0 −0.125 −0.0250.000 0.025 0.025 −0.025 0.000 0.025 0.000 112 0 −0.125 −0.025 0.0000.025 0.025 −0.025 0.025 0.025 0.000 113 0 −0.125 −0.025 0.025 0.0250.000 −0.025 0.025 0.025 −0.025 114 0 −0.125 −0.025 0.075 0.025 −0.0250.025 0.000 0.000 0.025 115 0 −0.125 0.000 0.050 0.025 0.000 −0.0250.025 0.025 −0.025 116 0 −0.125 0.000 0.075 0.025 −0.025 −0.025 0.0250.000 −0.025 117 0 −0.125 0.050 0.075 0.025 0.025 0.000 0.000 0.000−0.025 118 0 −0.125 0.075 0.075 −0.025 0.000 −0.025 −0.025 0.000 0.000119 0 −0.100 −0.075 −0.050 0.025 0.025 −0.025 −0.025 0.025 0.025 120 0−0.100 −0.050 −0.050 0.025 0.025 −0.025 −0.025 0.025 0.025 121 0 −0.100−0.050 −0.025 0.025 0.025 −0.025 −0.025 0.025 0.025 122 0 −0.100 −0.025−0.050 0.025 0.025 −0.025 −0.025 0.025 0.000 123 0 −0.100 −0.025 −0.0250.000 0.025 −0.025 −0.025 0.025 0.025 124 0 −0.100 −0.025 −0.025 0.0250.025 −0.025 −0.025 0.025 0.000 125 0 −0.100 0.050 0.075 −0.025 −0.025−0.025 −0.025 −0.025 0.000 126 0 −0.100 0.075 0.075 −0.025 0.000 −0.025−0.025 0.000 0.000 127 0 −0.100 0.075 0.075 0.000 0.000 −0.025 −0.025−0.025 −0.025 128 0 −0.100 0.075 0.075 0.000 0.000 −0.025 −0.025 0.000−0.025 129 0 −0.075 0.025 0.075 0.025 −0.025 −0.025 0.025 −0.025 −0.025130 0 −0.075 0.050 0.075 −0.025 −0.025 0.000 −0.025 0.000 0.025 131 0−0.075 0.050 0.075 −0.025 −0.025 0.025 0.000 0.025 0.025 132 0 −0.0750.050 0.075 0.025 −0.025 −0.025 0.000 −0.025 −0.025 133 0 −0.075 0.0500.075 0.025 0.000 −0.025 0.025 0.000 −0.025 134 0 −0.075 0.075 0.075−0.025 −0.025 −0.025 −0.025 0.000 0.000 135 0 −0.075 0.075 0.075 −0.025−0.025 −0.025 −0.025 0.000 0.025 136 0 −0.075 0.075 0.075 −0.025 −0.0250.000 −0.025 0.025 0.025 137 0 −0.075 0.075 0.075 −0.025 −0.025 0.0000.000 0.000 0.025 138 0 −0.075 0.075 0.075 −0.025 −0.025 0.025 0.0000.000 0.025 139 0 −0.075 0.075 0.075 −0.025 −0.025 0.025 0.000 0.0250.025 140 0 −0.050 −0.050 −0.075 0.025 0.025 −0.025 0.000 0.000 0.000141 0 −0.050 0.050 0.075 −0.025 −0.025 0.000 −0.025 0.000 0.025 142 0−0.050 0.050 0.075 −0.025 −0.025 0.000 −0.025 0.025 0.025 143 0 −0.0500.050 0.075 0.025 −0.025 −0.025 0.025 −0.025 −0.025 144 0 −0.050 0.0750.075 −0.025 −0.025 −0.025 −0.025 0.025 0.025 145 0 −0.050 0.075 0.075−0.025 −0.025 0.025 0.000 0.000 0.025 146 0 −0.050 0.075 0.075 −0.025−0.025 0.025 0.000 0.025 0.025 147 0 −0.025 0.075 0.075 −0.025 −0.0250.025 0.000 0.000 0.025 148 0 −0.025 0.075 0.075 −0.025 −0.025 0.0250.000 0.025 0.025 149 0 0.000 0.075 0.075 −0.025 −0.025 0.025 0.0000.000 0.025 150 0 0.000 0.075 0.075 −0.025 −0.025 0.025 0.000 0.0250.025 151 0 0.025 −0.050 −0.075 0.025 0.025 0.025 0.025 −0.025 −0.025152 0 0.050 0.075 −0.050 −0.025 0.025 −0.025 −0.025 −0.025 −0.025 153 00.075 0.075 −0.050 0.000 0.025 −0.025 −0.025 −0.025 −0.025 154 0 0.1000.050 −0.075 −0.025 0.000 −0.025 0.025 0.000 0.000 155 0 0.100 0.050−0.075 −0.025 0.025 0.000 0.025 0.000 −0.025 156 0 0.100 0.050 −0.075−0.025 0.025 0.025 0.025 0.025 0.000 157 0 0.100 0.050 −0.075 0.0000.025 0.000 0.000 −0.025 −0.025 158 0 0.100 0.075 −0.075 −0.025 0.000−0.025 0.000 0.000 0.000 159 0 0.100 0.075 −0.075 −0.025 0.025 0.0000.025 0.025 0.000 160 0 0.100 0.075 −0.075 −0.025 0.025 0.025 0.0250.025 0.025 161 0 0.125 0.050 −0.075 0.000 −0.025 −0.025 0.000 0.0000.000 162 0 0.125 0.075 −0.075 −0.025 0.000 −0.025 −0.025 0.000 0.000163 0 0.125 0.075 −0.075 −0.025 0.000 −0.025 0.000 0.000 0.000 164 00.125 0.075 −0.050 0.000 0.000 −0.025 0.000 −0.025 −0.025 165 0 0.1250.075 −0.050 0.000 0.000 −0.025 0.000 −0.025 0.000 166 0 0.125 0.075−0.050 0.000 0.000 −0.025 0.000 0.000 0.000 167 0 0.125 0.075 −0.0500.000 0.000 −0.025 0.000 0.025 0.025

18. Appendix D: Through Focus RIQ for Combinations of SphericalAberration in Appendix C

Combination −1.50 −1.25 −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.001.25 1.50 No Aberr 0.024 0.040 0.073 0.148 0.307 0.709 1.000 0.709 0.3070.148 0.073 0.040 0.024 101 0.071 0.102 0.206 0.371 0.466 0.446 0.4090.397 0.365 0.305 0.236 0.171 0.114 102 0.075 0.113 0.213 0.357 0.4210.407 0.430 0.459 0.402 0.301 0.220 0.160 0.110 103 0.071 0.106 0.2240.382 0.431 0.388 0.385 0.405 0.374 0.309 0.238 0.173 0.120 104 0.0450.079 0.216 0.430 0.524 0.446 0.376 0.385 0.383 0.326 0.240 0.161 0.106105 0.043 0.075 0.203 0.427 0.551 0.478 0.377 0.355 0.350 0.314 0.2420.160 0.101 106 0.045 0.108 0.230 0.382 0.459 0.413 0.366 0.386 0.3820.312 0.221 0.151 0.109 107 0.032 0.091 0.212 0.323 0.360 0.391 0.4630.483 0.407 0.317 0.255 0.198 0.141 108 0.044 0.109 0.239 0.330 0.3540.389 0.444 0.462 0.422 0.347 0.264 0.183 0.111 109 0.029 0.106 0.2310.314 0.358 0.427 0.489 0.478 0.403 0.321 0.251 0.176 0.107 110 0.0280.098 0.234 0.343 0.359 0.364 0.439 0.503 0.447 0.324 0.232 0.168 0.109111 0.033 0.093 0.221 0.343 0.385 0.402 0.469 0.514 0.446 0.326 0.2340.168 0.113 112 0.049 0.091 0.202 0.327 0.384 0.405 0.450 0.467 0.4000.303 0.223 0.163 0.116 113 0.048 0.082 0.211 0.400 0.476 0.408 0.3650.391 0.387 0.325 0.239 0.167 0.118 114 0.044 0.095 0.211 0.386 0.4860.426 0.358 0.375 0.370 0.305 0.231 0.167 0.119 115 0.053 0.096 0.2120.360 0.420 0.374 0.361 0.416 0.420 0.340 0.239 0.164 0.119 116 0.0670.121 0.220 0.342 0.392 0.355 0.361 0.434 0.455 0.389 0.277 0.169 0.101117 0.039 0.095 0.206 0.321 0.369 0.365 0.383 0.422 0.418 0.358 0.2680.180 0.120 118 0.061 0.120 0.212 0.315 0.388 0.387 0.350 0.353 0.3650.344 0.304 0.244 0.168 119 0.065 0.127 0.213 0.309 0.364 0.393 0.4320.436 0.395 0.342 0.269 0.183 0.111 120 0.040 0.098 0.211 0.322 0.3540.366 0.412 0.425 0.391 0.355 0.296 0.204 0.125 121 0.039 0.104 0.2360.352 0.374 0.383 0.441 0.469 0.426 0.351 0.264 0.173 0.102 122 0.0280.085 0.205 0.324 0.362 0.371 0.405 0.413 0.372 0.322 0.267 0.194 0.125123 0.039 0.083 0.201 0.313 0.367 0.431 0.486 0.458 0.392 0.348 0.2880.192 0.105 124 0.020 0.075 0.204 0.339 0.396 0.417 0.452 0.459 0.4030.317 0.242 0.172 0.107 125 0.044 0.096 0.203 0.327 0.395 0.383 0.3590.389 0.423 0.393 0.304 0.194 0.101 126 0.057 0.106 0.205 0.327 0.4100.411 0.368 0.358 0.369 0.346 0.293 0.224 0.147 127 0.038 0.087 0.2000.338 0.402 0.383 0.367 0.388 0.397 0.359 0.282 0.194 0.123 128 0.0370.097 0.206 0.319 0.378 0.380 0.379 0.396 0.381 0.319 0.250 0.188 0.134129 0.053 0.097 0.219 0.353 0.404 0.378 0.365 0.397 0.395 0.323 0.2350.163 0.112 130 0.050 0.106 0.211 0.342 0.446 0.474 0.421 0.381 0.3810.347 0.267 0.179 0.109 131 0.058 0.121 0.201 0.302 0.420 0.465 0.4190.397 0.393 0.330 0.238 0.161 0.104 132 0.025 0.082 0.215 0.346 0.3850.372 0.406 0.470 0.463 0.365 0.248 0.158 0.104 133 0.059 0.103 0.2050.318 0.370 0.369 0.394 0.451 0.437 0.328 0.219 0.151 0.109 134 0.0450.095 0.210 0.336 0.389 0.380 0.383 0.424 0.441 0.388 0.295 0.199 0.116135 0.046 0.094 0.209 0.331 0.379 0.374 0.371 0.392 0.413 0.383 0.3030.207 0.121 136 0.048 0.102 0.208 0.326 0.393 0.391 0.358 0.355 0.3770.356 0.289 0.213 0.142 137 0.028 0.082 0.201 0.325 0.378 0.368 0.3670.418 0.461 0.422 0.319 0.200 0.103 138 0.024 0.083 0.205 0.344 0.4240.411 0.371 0.380 0.404 0.376 0.299 0.206 0.126 139 0.036 0.107 0.2140.316 0.387 0.398 0.373 0.388 0.408 0.363 0.278 0.191 0.120 140 0.0670.117 0.201 0.311 0.384 0.416 0.461 0.485 0.422 0.312 0.219 0.151 0.102141 0.055 0.105 0.215 0.361 0.464 0.483 0.431 0.379 0.364 0.333 0.2560.169 0.101 142 0.075 0.131 0.218 0.317 0.399 0.438 0.415 0.382 0.3740.331 0.245 0.168 0.110 143 0.052 0.090 0.204 0.350 0.411 0.382 0.3710.406 0.398 0.313 0.222 0.161 0.118 144 0.078 0.118 0.208 0.319 0.3810.398 0.405 0.407 0.399 0.353 0.273 0.194 0.124 145 0.028 0.086 0.2120.359 0.437 0.421 0.381 0.386 0.403 0.368 0.286 0.192 0.116 146 0.0360.105 0.226 0.341 0.402 0.405 0.382 0.390 0.405 0.360 0.269 0.179 0.109147 0.035 0.092 0.218 0.372 0.454 0.434 0.387 0.383 0.391 0.352 0.2720.183 0.111 148 0.042 0.104 0.231 0.363 0.423 0.415 0.388 0.386 0.3920.348 0.260 0.171 0.104 149 0.046 0.102 0.223 0.381 0.471 0.449 0.3910.374 0.371 0.329 0.255 0.177 0.110 150 0.053 0.107 0.230 0.378 0.4490.430 0.391 0.375 0.370 0.328 0.249 0.168 0.104 151 0.087 0.139 0.2180.318 0.389 0.428 0.447 0.425 0.379 0.315 0.228 0.150 0.103 152 0.0480.099 0.206 0.320 0.374 0.384 0.417 0.463 0.443 0.336 0.220 0.154 0.125153 0.042 0.095 0.205 0.324 0.375 0.387 0.427 0.466 0.430 0.318 0.2090.153 0.130 154 0.075 0.124 0.201 0.316 0.436 0.454 0.387 0.368 0.3670.303 0.217 0.152 0.104 155 0.072 0.118 0.205 0.348 0.488 0.481 0.3760.359 0.381 0.320 0.222 0.157 0.118 156 0.040 0.096 0.200 0.357 0.5040.508 0.407 0.366 0.363 0.301 0.213 0.155 0.119 157 0.047 0.097 0.2020.355 0.455 0.420 0.357 0.393 0.426 0.345 0.223 0.156 0.132 158 0.0530.110 0.206 0.316 0.403 0.413 0.369 0.385 0.428 0.385 0.276 0.183 0.122159 0.071 0.127 0.209 0.315 0.415 0.418 0.355 0.370 0.417 0.368 0.2600.175 0.126 160 0.050 0.107 0.206 0.329 0.429 0.429 0.363 0.363 0.3890.335 0.236 0.164 0.125 161 0.056 0.121 0.211 0.304 0.386 0.420 0.4000.393 0.387 0.319 0.226 0.161 0.121 162 0.055 0.122 0.222 0.313 0.3550.361 0.363 0.401 0.449 0.410 0.285 0.170 0.107 163 0.063 0.129 0.2330.335 0.403 0.411 0.363 0.354 0.400 0.387 0.291 0.189 0.118 164 0.0620.106 0.202 0.330 0.412 0.421 0.394 0.375 0.371 0.348 0.275 0.177 0.105165 0.050 0.107 0.217 0.345 0.423 0.426 0.379 0.351 0.361 0.332 0.2400.151 0.101 166 0.047 0.105 0.201 0.312 0.411 0.459 0.438 0.418 0.4200.366 0.262 0.173 0.112 167 0.053 0.119 0.210 0.307 0.405 0.466 0.4470.416 0.394 0.311 0.212 0.161 0.122

The invention claimed is:
 1. A lens for an eye, the lens having anoptical axis and an aberration profile about its optical axis, theaberration profile: having a focal distance; and including higher orderaberrations having at least four spherical aberration componentsincluding at least one of a primary spherical aberration component(C(4,0)) and a secondary spherical aberration component (C(6,0)),wherein the aberration profile provides, for a model eye with noaberrations and an on-axis length equal to the focal distance: a retinalimage quality (RIQ) with a through focus slope that degrades in adirection of eye growth; and a RIQ of at least 0.30; wherein said RIQ isVisual Strehl Ratio measured along the optical axis for at least onepupil diameter in the range 3 mm to 6 mm, over a spatial frequency rangeof 0 to 30 cycles/degree inclusive and at a wavelength selected fromwithin the range 540 nm to 590 nm inclusive; and wherein the aberrationprofile provides a RIQ of at least 0.30 at the focal length for allpupil diameters in the range 3 mm to 6 mm.
 2. The lens of claim 1,wherein the higher order aberrations include at least two sphericalaberration terms selected from the group C(8,20) to C(20,0).
 3. The lensof claim 1, wherein the average slope over a horizontal field of atleast −20° to +20° degrades in a direction of eye growth.
 4. The lens ofclaim 1, wherein the aberration profile provides a RIQ with a throughfocus slope that degrades in a direction of eye growth when primaryastigmatism is added to the aberration profile.
 5. The lens of claim 1,wherein the aberration profile provides a RIQ with a through focus slopethat degrades in a direction of eye growth when secondary astigmatism isadded to the aberration profile.
 6. The lens of claim 1, wherein saidRIQ is${{RIQ} = \frac{\int{\int_{- {Fmin}}^{+ {Fmax}}{{{CSF}\left( {x,y} \right)}*\left( \left( \left( {{FT}\left( {❘{{FT}\left\{ {{A\left( {\rho,\theta} \right)} \star {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {W\left( {\rho,\theta} \right)}} \right\rbrack}} \right\}}❘}^{2} \right)} \right) \right) \right)}}}{\int{\int_{- {Fmin}}^{+ {Fmax}}{{{CSF}\left( {x,y} \right)}*\left( \left( \left( {{FT}\left( {❘{{FT}\left\{ {{A\left( {\rho,\theta} \right)} \star {\exp\left\lbrack {\frac{2\pi i}{\lambda} \star {{Wdiff}\left( {\rho,\theta} \right)}} \right\rbrack}} \right\}}❘}^{2} \right)} \right) \right) \right)}}}},$wherein: Fmin is 0 cycles/degree and Fmax is 30 cycles/degree; CSF(x, y)denotes the contrast sensitivity functionCSF(F)=2.6(0.0192+0.114f)e^(−(0.114f){circumflex over ( )}1.1), where fspecifies the tested spatial frequency, in the range of F_(min) toF_(max); FT denotes a 2D fast Fourier transform; A(ρ, θ) denotes thepupil diameter; W(ρ, θ) denotes wavefront phase of the test case${W\left( {\rho,\theta} \right)} = {\sum_{i = 1}^{k}{a_{i}{Z_{i}\left( {\rho,\theta} \right)}}}$measured for i=1 to 20; Wdiff(ρ, θ) denotes wavefront phase of thediffraction limited case; ρ and θ are normalised polar coordinates,where p represents the radial coordinate and θ represents the angularcoordinate or azimuth; and λ denotes wavelength.